
619 F.3d 187 (2010)
John MANNIX, Petitioner-Appellant,
v.
William PHILLIPS, Respondent-Appellee.
Jermaine Archer, Petitioner-Appellant,
v.
Brian Fischer, Respondent-Appellee.
Docket Nos. 07-0664-pr, 09-2041-pr.
United States Court of Appeals, Second Circuit.
Argued: July 12, 2010.
Decided: August 30, 2010.
*188 Randa D. Maher, Great Neck, New York, for Petitioner-Appellant John Mannix.
*189 Sheryl Feldman, Assistant District Attorney, for Cyrus R. Vance, Jr., District Attorney, New York County, New York, NY, for Respondent-Appellee William Phillips.
Sally Wasserman, New York, NY, for Petitioner-Appellant Jermaine Archer.
Howard B. Goodman (Leonard Joblove, Sholom J. Twersky, Assistant District Attorneys, on the brief), Assistant District Attorney, for Charles J. Hynes, District Attorney, Kings County, Brooklyn, NY, for Respondent-Appellee Brian Fischer.
Before: RAGGI, WESLEY, HALL, Circuit Judges.
WESLEY, Circuit Judge:
John Mannix appeals from a judgment of the United States District Court for the Southern District of New York (Richard Conway Casey, Judge) denying his petition for a writ of habeas corpus. Jermaine Archer appeals from a judgment of the United States District Court for the Eastern District of New York (Joseph F. Bianco, Judge) denying his petition for a writ of habeas corpus. We heard the appeals in tandem because both petitioners argue that, at the time their convictions became final, the definition of depraved indifference murder under New York law, see N.Y. Penal Law § 125.25(2), was unconstitutionally vague. Mannix further argues that the evidence was legally insufficient to support his conviction.[1]
The New York state courts' rejection of petitioners' vagueness claims was not contrary to, or an unreasonable application of, clearly established Supreme Court precedent. 28 U.S.C. § 2254(d)(1). At the times petitioners' convictions became final, New York's murder statute, N.Y. Penal Law § 125 with notice that their conduct was prohibited. And, the statute did not authorize or encourage arbitrary enforcement.
Petitioners argue that, at the time that each of their convictions became final, the depraved indifference murder statute, id. § 125.25(2), was constitutionally defective because the differences between that statute and the statute governing reckless manslaughter, or manslaughter in the second degree, id. § 125.15(1), were indiscernible or nonexistent. In support, petitioners cite two decisions by a former judge of the United States District Court for the Southern District of New York granting habeas corpus relief from convictions on this ground. See St. Helen v. Senkowski, No. 02 Civ. 10248, 2003 WL 25719647 (S.D.N.Y. Sept.22, 2003), rev'd on other grounds, 374 F.3d 181 (2d Cir.2004); Jones v. Keane, No. 02 Civ. 1804, 2002 WL 33985141 (S.D.N.Y. May 22, 2002), rev'd on other grounds, 329 F.3d 290 (2d Cir.2003). We conclude that the argument lacks merit. The depraved indifference murder and reckless manslaughter statutes were distinct at the times that petitioners' convictions were finalized. And, in any event, an act may violate more than one criminal statute, and the government may prosecute a defendant under either one without raising constitutional vagueness concerns under the Due Process Clause. What the Constitution's Equal Protection Clause demands is that the government not discriminate against a class of defendants in the exercise of its prosecutorial discretion.
We also hold that legally sufficient evidence supports Mannix's conviction. Finally, we need not resolve the question of *190 whether or not, pursuant to the prison mailbox rule, Mannix's objections to the magistrate judge's report and recommendation are properly presented to this Court. Even if this claim is properly before us, Mannix is not entitled to any relief on this basis because the magistrate's report and recommendation were not in error.

I. BACKGROUND

A. Petitioner John Mannix
In the early morning hours of February 26, 2000, in a Manhattan bar, Mannix got into a heated exchange with an individual named Matthew Torruella. Mannix v. Phillips, 390 F.Supp.2d 280, 283 (S.D.N.Y. 2005). Eventually, the two men ended up in a physical altercation. Id. The dispute appeared to come to an end when Mannix pinned Torruella up against a wall and the two men then separated. Id. However, as Mannix was backing away, Torruella "sucker-punched" him in the face. Id. Apparently recognizing that Mannix was angry, Torruella's companion pulled Torruella into the ladies' room at the back of the bara one-person restroom, approximately four feet by eight feet in sizeand locked the door. Id. Moments later, Mannix followed. Id.
According to witnesses, Mannix kicked and pounded the bathroom door for between thirty seconds and two minutes. Id. At this point, as least one witness indicated that she heard something that sounded like a gunshot while Mannix was still in front of the door. Id. at 283-84. Others testified to hearing loud pounding noises and something that sounded like someone throwing himself against a door. Id. Mannix fatally shot Torruella in the chest through the bathroom door. Id. at 284. Mannix then left, but later called the bar and asked if he had "hit anyone." Id. When he was told that he had, he responded, "good," and hung up. Id. An investigating officer determined that the bullet hole in the bathroom door was "dead center in the middle of the door at approximately chest level." Id. (internal quotation marks omitted).
Mannix was charged with depraved indifference murder, N.Y. Penal Law § 125.25(2), and the lesser included offense of reckless manslaughter, id. § 125.15(1). The court explained to the jury that, if it was "satisfied beyond a reasonable doubt of the guilt of the defendant," it could "find him guilty of only one of these crimes." Trial Tr. 1067:17-19. The judge informed the jury that a person is guilty of depraved indifference murder "when under circumstances evincing a depraved indifference to human life he recklessly engages in conduct which creates a grave risk of death to another person, and thereby causes the death of that person."[2] Trial Tr. 1068:9-12. The court made it clear that in order for the jury to find Mannix guilty of depraved indifference murder, it had to "decide whether the circumstances surrounding his reckless conduct when objectively viewed, made it so uncaring, so callous, so dangerous and so inhuman as to demonstrate *191 an attitude of total and utter disregard for the life of the person or persons endangered." Trial Tr. 1069:17-21. The judge further instructed:
Under our law a crime committed recklessly is generally regarded as less serious and blameworthy than a crime committed intentionally. But when reckless conduct is engaged in under circumstances evincing a depraved indifference to human life, the law regards that conduct as so serious, so egregious as to be the equivalent of intentional conduct.
Conduct evincing a depraved indifference to human life is much more serious and blameworthy than conduct which is merely reckless. It is conduct which beyond being reckless is so wanton, so deficient of moral sense and concern, so devoid of regard for life or lives of others, as to equal in blameworthiness intentional conduct, which produces the same result. Trial Tr. 1069:1-14.
The judge instructed that if the jury did not find Mannix guilty of depraved indifference murder, it should next consider whether the government proved beyond a reasonable doubt that he was guilty of reckless manslaughter. The judge explained that "a person is guilty of [reckless] manslaughter ... when that person recklessly causes the death of another person." Trial Tr. 1070:17-19. On November 13, 2000, the jury found Mannix guilty of depraved indifference murder.[3] N.Y. Penal Law § 125.25(2).
Mannix appealed, and the appellate division affirmed his conviction. People v. Mannix, 302 A.D.2d 297, 756 N.Y.S.2d 33, 34 (1st Dep't 2003). The appellate division held that, "[c]ontrary to [Mannix's] contention, Penal Law § 125.25(2), which defines `depraved indifference' murder, is not unconstitutionally vague." Id. The court stated that depraved indifference murder and reckless manslaughter were "separate crimes, both facially and as interpreted." Id. The appellate division further noted that, "[i]n any event, the Supreme Court of the United States `has long recognized that when an act violates more than one criminal statute, the [g]overnment may prosecute under either so long as it does not discriminate against any class of defendants.'" Id. (quoting United States v. Batchelder, 442 U.S. 114, 123-24, 99 S.Ct. 2198, 60 L.Ed.2d 755 (1979)). The appellate division also found that "[t]he verdict convicting [Mannix] of depraved indifference murder was based on legally sufficient evidence and was not against the weight of the evidence." Id. The New York Court of Appeals denied Mannix's application for leave to appeal. People v. Mannix, 100 N.Y.2d 622, 767 N.Y.S.2d 406, 799 N.E.2d 629 (2003).
Mannix filed a pro se petition for a writ of habeas corpus in the Southern District of New York. Mannix, 390 F.Supp.2d at 282. Judge Casey referred the matter to Magistrate Judge Gorenstein, to whom Mannix argued that "the difference between depraved indifference murder and reckless manslaughter is no longer `well understood' in New York," that the depraved indifference murder statute "does not prohibit the proscribed conduct `with sufficient definiteness,'" and that, therefore, the depraved indifference murder statute is unconstitutionally vague. Id. at 288. Mannix also maintained that his conviction for depraved indifference murder was not supported by legally sufficient evidence. Id. at 293-295.
In a report and recommendation dated May 25, 2005, the magistrate rejected Mannix's argument that the alleged congruence *192 between the depraved indifference murder and the reckless manslaughter statutes rendered his conviction constitutionally infirm. Id. at 292. The judge reasoned that, "[e]ven assuming arguendo that there is no distinction at all between the conduct covered by the depraved indifference murder statute and the conduct covered by the manslaughter statute, there is no clearly established federal constitutional principles permitting [the district court] to grant habeas relief." Id. The court further held that the appellate division's conclusion that legally sufficient evidence supported Mannix's conviction was "not an unreasonable application of Supreme Court law." Id. at 295. Accordingly, the magistrate recommended that the petition for habeas corpus be denied.
The magistrate judge provided notice to the parties that they had ten days during which they could object to his report. Id. at 296; see also id. at 282. No objections were filed within that period. The district court, finding no clear error, accepted the magistrate's report and recommendation. Id. at 282.
On May 18, 2007, Mannix filed a motion for a certificate of appealability in this Court, and included a document entitled "Petitioner's Objections to the Magistrate's Report and Recommendation." The document, and an accompanying certificate of servicenever docketed in the district courtare dated June 3, 2005. Had Mannix's objections been received by the district court on that date, they would have been considered timely.
On July 9, 2009, a panel of this Court granted, in part, Mannix's motion for a certificate of appealability and certified three questions for appeal: (1) whether the depraved indifference murder statute was unconstitutionally vague at the time petitioner's conviction became final; (2) whether petitioner properly objected to the magistrate's report when the district court did not refer to his objections, but where petitioner attached a copy of his objections to his certificate of appealability papers; and (3) whether petitioner's conviction rests on legally sufficient evidence.[4]

B. Petitioner Jermaine Archer
On July 20 and 21, 1997, Petitioner Jermaine Archer had several brief street confrontations with Carlos Bethune and Reynaldo Niles. On the night of July 21, 1997, Bethune was driving his car with Reynaldo's brother, Patrick Niles, in the passenger seat. At one point, Bethune slowed down and noticed a van to the right of his car. According to Bethune, Archer then emerged from around the van, pulled a gun from his waist, and fired three or four shots, in quick succession, into Bethune's car before Bethune could drive away. One of these shots struck Patrick Niles in the back of the neck and killed him. Archer v. Fischer, No. 05 Civ. 4990(JFB), 2009 WL 1011591, at *1-2 (E.D.N.Y. Apr.13, 2009).
Archer was charged with intentional murder, N.Y. Penal Law § 125.25(1), depraved indifference murder, id. § 125.25(2), manslaughter in the first degree, id. § 125.20(1), and manslaughter in the second degree, also known as reckless manslaughter, id. § 125.15(1). Archer was tried before a jury. At the close of the trial, the judge instructed the jury that it should consider the intentional murder charge first, and consider the offense of depraved indifference murder only if it found petitioner Archer not guilty of intentional murder. Similarly, the judge instructed *193 that the jury should consider the charge of manslaughter in the first degree only if it concluded Archer was not guilty of depraved indifference murder, and it should consider the reckless manslaughter charge last and only if it found Archer not guilty of all previously considered charges.
With respect to the depraved indifference murder charge, the court explained:
A person is guilty of depraved indifference murder in the second degree when, under circumstances evincing a depraved indifference to human life, he recklessly creates a grave and unjustifiable risk of death to another person and thereby causes that person's death.
A grave risk of death means a very substantial, very great risk of death. A person acts recklessly regarding a grave and unjustifiable risk of death when he is aware of and consciously disregards a very substantial and unjustifiable risk that death will result from his actions. The risk must be of such [a] nature and degree that the defendant's disregard of it was a gross deviation, meaning a very great, flagrant, obvious departure from the standard of conduct that a reasonable person would have followed in the same situation.
Although this charge is called murder, it does not require proof of the defendant's intent to kill or even intent to murder anyone. In depraved indifference murder, taking the place of an intent to kill are the defendant's creation of a very great and unjustifiable risk of death, plus his reckless state of mind regarding that risk, plus circumstances evincing a depraved indifference to human life.
Trial Tr. 516:2-25, 517:1. In further explicating the meaning of depraved indifference, the court stated that "it relates to the dangerous circumstances alleged to have been created by the defendant." Trial Tr. 518:24-25, 519:1. The court noted that "[a] depraved indifference to human life is much more serious and blameworthy than conduct that is merely reckless, because the conduct is much more dangerous." Trial Tr. 519:2-5. The court instructed:
Circumstances evincing a depraved indifference to human life exist[ ] when, in the judgment of the jury, beyond a reasonable doubt, the defendant's conduct, beyond being reckless, was so highly and immediately dangerous to life and so wanton, without regard for or caring about the results to human life, and deficient in moral sense or concern, that it warrants the same level of criminal blame as the law imposes on someone who intentionally causes a person's death even though there is no intent to kill.
Trial Tr. 519:5-14. Later in the instructions provided to the jury, the court explained the charge of reckless manslaughter. The judge stated:
A person is guilty of manslaughter in the second degree when he recklessly causes another person's death. A person acts recklessly regarding someone's death in manslaughter in the second degree when he is aware of and consciously disregards a substantial and unjustifiable risk of death from his actions, and the risk is of such nature and degree that disregarding it is a gross deviation from the standard of conduct of a reasonable person.
Trial Tr. 521:24-25, 522:1-7. The court then explicitly contrasted reckless manslaughter with depraved indifference murder. The court acknowledged that the crimes might be viewed as "similar" in that they both required proof that the defendant recklessly caused someone's death. Trial Tr. 522:14. However, the court made it clear to the jury that the crimes are *194 distinct because "manslaughter in the second degree does not require proof of circumstances evincing a depraved indifference... to human life." Trial Tr. 522:17-19.
Archer was convicted of depraved indifference murder.[5] N.Y. Penal Law § 125.25(2). Archer appealed his conviction to the appellate division.[6] While his appeal to the appellate division was pending, Archer made a motion in state supreme court, pursuant to New York Criminal Procedure Law § 440.10, to vacate the judgment of conviction.[7] One of the grounds for this motion was Archer's contention that the depraved indifference murder statute was unconstitutionally vague. The state trial court denied his § 440.10 motion, holding that the depraved indifference murder statute was not unconstitutionally vague. Petitioner applied for leave to appeal the denial of his § 440.10 motion; this motion was denied by the appellate division. On October 25, 2004, the appellate division affirmed Archer's conviction. People v. Archer, 11 A.D.3d 704, 784 N.Y.S.2d 567, 567 (2d Dep't 2004).[8] Archer sought leave to appeal to the New York Court of Appeals, which was denied. People v. Archer, 4 N.Y.3d 741, 790 N.Y.S.2d 653, 824 N.E.2d 54 (2004).
Archer then petitioned for a writ of habeas corpus in the Eastern District of New York. Archer, 2009 WL 1011591, at *5. The court held that Archer's vagueness challenge to New York's depraved indifference murder statute was without merit. Id. at *24-26. The court concluded that the "statute gives the accused fair warning that his conduct was criminal and proscribed since ordinary people would understand that shooting at someone at short range in the vicinity of others would be criminal and put them at risk of a homicide conviction." Id. at *25 (internal quotation marks omitted). The court further reasoned that the language of the statute was adequate to prevent arbitrary and discriminatory enforcement. Id. at *26. The district judge determined that "the jury instructions administered in this case gave the jury sufficient guidelines to prevent arbitrary or discriminatory application of the statute." Id.
On October 13, 2009, this Court granted, in part, Archer's motion for a certificate of appealability. The certificate issued by this Court was limited to the issue of whether New York's depraved indifference murder statute was unconstitutionally vague as compared to New York's reckless *195 manslaughter statute at the time Archer's conviction became final.[9]

II. STANDARD OF REVIEW
The Antiterrorism and Effective Death Penalty Act of 1996 ("AEDPA") provides that persons in state custody because of a state court conviction may petition for federal habeas corpus relief if their custody is "in violation of the Constitution or laws or treaties of the United States."[10] 28 U.S.C. § 2254(a); see Hawkins v. Costello, 460 F.3d 238, 242 (2d Cir.2006). We review de novo the district courts' decisions to deny the petitions for habeas corpus. Jenkins v. Artuz, 294 F.3d 284, 290 (2d Cir.2002). However, we review the district courts' factual findings only for clear error.[11]Id.
In order to obtain relief, the petitioners must show that the resolutions of the merits of their claims in the state courts were "contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States," or that they were "based on an unreasonable determination of the facts in light of the evidence presented in the State court proceeding[s]."[12] 28 U.S.C. § 2254(d)(1)-(2). A decision is contrary to Supreme Court precedent if the state court "applie[d] a rule that contradicts" that precedent, or reached a different result than the Supreme Court on facts that are "materially indistinguishable." Williams v. Taylor, 529 U.S. 362, 405-06, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). We are charged with evaluating whether the state courts' applications of clearly established federal law were "objectively unreasonable." Id. at 409, 120 S.Ct. 1495. Applying these standards, we conclude that neither Mannix nor Archer is entitled to relief from his state court conviction.

III. DISCUSSION

A. Threshold Procedural Requirements
Respondents submit that we should not address the merits of petitioners' vagueness *196 challenges to New York's depraved indifference murder statute for procedural reasons. As to Mannix, respondent argues that he waived his habeas claims by failing to file timely objections to the magistrate judge's report and recommendation in the district court. See Small v. Sec'y of Health & Human Servs., 892 F.2d 15, 16 (2d Cir.1989); see also Thomas v. Arn, 474 U.S. 140, 155, 106 S.Ct. 466, 88 L.Ed.2d 435 (1985). Mannix concedes that his objections to the magistrate's report were never received by the district court. And, it is clear from the record that the district court was not on notice of Mannix's objections at the time it accepted the report and recommendation of the magistrate judge. Mannix, 390 F.Supp.2d at 282 (noting that "[n]o objections ha[d] been filed"). Nonetheless, Mannix contends that his objections were timely filed under the prison mailbox rule because he delivered them to prison officials on June 3, 2005. See Houston v. Lack, 487 U.S. 266, 270, 108 S.Ct. 2379, 101 L.Ed.2d 245 (1988); Fed. R.App. P. 4(c)(1).
We need not specifically decide whether the prison mailbox rule applies in the context of objections to a report and recommendation, cf. Noble v. Kelly, 246 F.3d 93, 97 (2d Cir.2001), or whether Mannix's objections were waived through lack of diligence on his part, compare Huizar v. Carey, 273 F.3d 1220, 1223 (9th Cir.2001), with Allen v. Culliver, 471 F.3d 1196, 1198 (11th Cir.2006), because the waiver rule is "nonjurisdictional," Spence v. Superintendent, Great Meadow Corr. Facility, 219 F.3d 162, 174 (2d Cir.2000), and we conclude, in any event, that Mannix's claims fail on the merits.
As to Archer, respondent asserts that his claim should be deemed exhausted and procedurally defaulted. See, e.g., Coleman v. Thompson, 501 U.S. 722, 729-30, 111 S.Ct. 2546, 115 L.Ed.2d 640 (1991). Archer maintains that the state waived its procedural default defense by failing to raise it in the district court. Before AEDPA, it was well established that "procedural default is normally a defense that the State is obligated to raise and preserve if it is not to lose the right to assert the defense thereafter." Trest v. Cain, 522 U.S. 87, 89, 118 S.Ct. 478, 139 L.Ed.2d 444 (1997) (internal quotation marks and brackets omitted). Under AEDPA, however, "[a] State shall not be deemed to have waived the exhaustion requirement or be estopped from reliance upon the requirement unless the State, through counsel, expressly waives the requirement." 28 U.S.C. § 2254(b)(3); see also Lurie v. Wittner, 228 F.3d 113, 123 (2d Cir.2000).
Again, we need not decide whether AEDPA disfavors a state's waiver of its procedural default defense when that default is based on petitioner's failure properly to exhaust, compare Franklin v. Johnson, 290 F.3d 1223, 1231 (9th Cir. 2002), with McNair v. Campbell, 416 F.3d 1291, 1305-06, n. 13 (11th Cir.2005), because, Archer's claim, like Mannix's, fails in any event on the merits, see 28 U.S.C. § 2254(b)(2); see also Zarvela v. Artuz, 364 F.3d 415, 417 (2d Cir.2004).

B. Petitioners' Vagueness Challenges to their Convictions for Depraved Indifference Murder
Both Mannix and Archer contend that their convictions for depraved indifference murder should be vacated on the ground that the statute under which they were convicted was void for vagueness. Petitioners cannot demonstrate that the state courts unreasonably applied clearly established federal law in concluding that New York's depraved indifference murder statute was not void for vagueness as applied to them. 28 U.S.C. § 2254(d).


*197 i. The Vagueness Doctrine
The void-for-vagueness doctrine derives from the constitutional guarantee of due process, which requires that a penal statute define a criminal offense "`[1] with sufficient definiteness that ordinary people can understand what conduct is prohibited and [2] in a manner that does not encourage arbitrary and discriminatory enforcement.'" Skilling v. United States, ___ U.S. ___, 130 S.Ct. 2896, 2927-28, 177 L.Ed.2d 619 (2010) (alterations in original) (quoting Kolender v. Lawson, 461 U.S. 352, 357, 103 S.Ct. 1855, 75 L.Ed.2d 903 (1983)). The "touchstone" of the notice prong "is whether the statute, either standing alone or as construed, made it reasonably clear at the relevant time that the defendant's conduct was criminal." United States v. Lanier, 520 U.S. 259, 267, 117 S.Ct. 1219, 137 L.Ed.2d 432 (1997). The arbitrary enforcement prong requires that a statute give "minimal guidelines" to law enforcement authorities, so as not to "permit a standardless sweep that allows policemen, prosecutors, and juries to pursue their personal predilections." Kolender, 461 U.S. at 358, 103 S.Ct. 1855 (internal quotation marks and brackets omitted). Although a law must provide "explicit standards," it "need not achieve meticulous specificity, which would come at the cost of flexibility and reasonable breadth." Dickerson v. Napolitano, 604 F.3d 732, 747 (2d Cir.2010) (internal quotation marks omitted).
The Supreme Court has made it clear that when the law "unambiguously specif[ies] the activity proscribed," the fact that this proscribed conduct may violate more than one statute does not render the statute void for vagueness. United States v. Batchelder, 442 U.S. 114, 123, 99 S.Ct. 2198, 60 L.Ed.2d 755 (1979). To the contrary, "when an act violates more than one criminal statute, the [g]overnment may prosecute under either so long as it does not discriminate against any class of defendants." Id. at 123-24, 99 S.Ct. 2198.
When, as here, the challenged law does not threaten First Amendment interests, we generally evaluate a vagueness claim only as applied to the facts of the particular case. See Maynard v. Cartwright, 486 U.S. 356, 361, 108 S.Ct. 1853, 100 L.Ed.2d 372 (1988); United States v. Nadi, 996 F.2d 548, 550 (2d Cir.1993); see also Dickerson, 604 F.3d at 743-45. Because "[a] plaintiff who engages in some conduct that is clearly proscribed cannot complain of the vagueness of the law as applied to the conduct of others," Vill. of Hoffman Estates v. Flipside, Hoffman Estates, Inc., 455 U.S. 489, 495, 102 S.Ct. 1186, 71 L.Ed.2d 362 (1982), we will uphold a statute against an as-applied challenge if "`the particular enforcement at issue [is] consistent with the core concerns underlying the [statute],'" Dickerson, 604 F.3d at 748 (alterations in original) (quoting Farrell v. Burke, 449 F.3d 470, 493 (2d Cir. 2006)). Put another way, "[o]ne to whose conduct a statute clearly applies may not successfully challenge it for vagueness." Parker v. Levy, 417 U.S. 733, 756, 94 S.Ct. 2547, 41 L.Ed.2d 439 (1974).

ii. New York's Depraved Indifference Murder Statute
New York's depraved indifference murder statute provides that a person is guilty of murder in the second degree when: "Under circumstances evincing a depraved indifference to human life, he recklessly engages in conduct which creates a grave risk of death to another person, and thereby causes the death of another person." N.Y. Penal Law § 125.25(2). At the times of petitioners' trials, and at the times that each of their convictions became final, recklessness was the required mental state for depraved indifference murder. People *198 v. Register, 60 N.Y.2d 270, 274, 469 N.Y.S.2d 599, 457 N.E.2d 704 (1983). Depravity and indifference were assessed by the jury, based on its review of the circumstances of the crime. Id. at 274-75, 469 N.Y.S.2d 599, 457 N.E.2d 704. In other words, a jury was required to evaluate the facts attendant to the crime, and the defendant's behavior in committing the crime, to make a determination as to whether those facts and behavior evinced a depraved indifference to human life. As the New York Court of Appeals explained, it was within the jury's province "to make a qualitative judgment whether defendant's act was of such gravity that it placed the crime upon the same level as the taking of life by premeditated design. It had to determine from the evidence if defendant's conduct, though reckless, was equal in blameworthiness to intentional murder." Id.
During the relevant time period, "depraved indifference" referred to "objective circumstances" surrounding the offenses committed. Id. at 276, 469 N.Y.S.2d 599, 457 N.E.2d 704. And, the definition of depraved indifference murder was "well understood." Id. at 279, 469 N.Y.S.2d 599, 457 N.E.2d 704 (internal quotation marks omitted); see also People v. Johnson, 87 N.Y.2d 357, 361, 639 N.Y.S.2d 776, 662 N.E.2d 1066 (1996); People v. Cole, 85 N.Y.2d 990, 992, 629 N.Y.S.2d 166, 652 N.E.2d 912 (1995). In Register, which governs petitioners' appeals, the New York Court of Appeals explained that depraved indifference murder "is not and never has been considered as a substitute for intentional homicide." 60 N.Y.2d at 279, 469 N.Y.S.2d 599, 457 N.E.2d 704. Moreover, it reasoned, depraved indifference murder "is distinguishable from manslaughter ... by the objective circumstances in which the act occurs." Id. at 278, 469 N.Y.S.2d 599, 457 N.E.2d 704.
Subsequent to petitioners' convictions becoming final, this interpretation of the depraved indifference murder statute changed. The New York Court of Appeals overturned its prior precedent and held that "depraved indifference to human life is a culpable mental state." People v. Feingold, 7 N.Y.3d 288, 294, 819 N.Y.S.2d 691, 852 N.E.2d 1163 (2006). As now understood by the New York Court of Appeals, "`a one-on-one shooting or knifing (or similar killing) can almost never qualify as depraved indifference murder.'" Policano v. Herbert, 7 N.Y.3d 588, 601, 825 N.Y.S.2d 678, 859 N.E.2d 484 (2006) (quoting People v. Payne, 3 N.Y.3d 266, 272, 786 N.Y.S.2d 116, 819 N.E.2d 634 (2004)). But see People v. Sanchez, 98 N.Y.2d 373, 383, 748 N.Y.S.2d 312, 777 N.E.2d 204 (2002) (observing that requirements that conduct causing death be "aimed at no one in particular" and must "endanger indiscriminately the lives of many" were "entirely obsolete" under then-existing New York law (internal quotation marks and alteration omitted)). Contrary to the assertions of petitioners, this change in construction is of little import in evaluating these petitions for habeas corpus relief based on vagueness, for on habeas review, we must look to New York law as it existed when Mannix and Archer's convictions became final. See Henry v. Ricks, 578 F.3d 134, 141 (2d Cir.2009); see also Policano v. Herbert, 453 F.3d 79, 83 (2d Cir.2006)(Raggi, J., dissenting from denial of rehearing en banc).

iii. Petitioners' Vagueness Challenges Lack Merit
We have no difficulty concluding that Mannix and Archer were both on notice that their conductshooting into an enclosed space when each petitioner knew people were insidewas criminal. Grayned v. City of Rockford, 408 U.S. 104, 108, 92 S.Ct. 2294, 33 L.Ed.2d 222 (1972). *199 It cannot be said that the depraved indifference murder statute, as interpreted at the relevant time, could "trap the innocent by not providing fair warning." Id. And, it certainly cannot be said that there was any risk that either man was unaware that his conduct was proscribed, such that their vagueness challenges would survive as-applied review. See Vill. of Hoffman Estates, 455 U.S. at 495, 102 S.Ct. 1186; accord Nadi, 996 F.2d at 550.
Through the time petitioners' convictions became final, New York courts consistently held that firing into a crowd or enclosed space is a "quintessential" case of depraved indifference murder. See People v. Suarez, 6 N.Y.3d 202, 214, 811 N.Y.S.2d 267, 844 N.E.2d 721 (2005) (observing that "[q]uintessential examples" of depraved indifference murder include "firing into a crowd" (citing People v. Jernatowski, 238 N.Y. 188, 192, 144 N.E. 497 (1924) (upholding depraved indifference murder conviction where "defendant fired two or more shots into the house where he knew there were human beings"))); accord Payne, 3 N.Y.3d at 271, 786 N.Y.S.2d 116, 819 N.E.2d 634; People v. Gonzalez, 1 N.Y.3d 464, 467, 775 N.Y.S.2d 224, 807 N.E.2d 273 (2004); see also People v. Brown, 173 A.D.2d 318, 575 N.Y.S.2d 460, 460 (1st Dep't 1991) (upholding depraved indifference murder conviction where defendant fired shot through closed door).
Relying on two decisions from the Southern District of New York, Mannix and Archer argue that the depraved indifference murder statute, as applied to them, encouraged arbitrary enforcement because their conduct was indistinguishable from conduct proscribed by the reckless manslaughter statute. See St. Helen, 2003 WL 25719647, at *8; Jones, 2002 WL 33985141, at *5. At the outset, we reject the premise of petitioners' argument. Under Register, depraved indifference murder was defined as distinct from, and requiring an element in addition to, the lesser included offense of reckless manslaughter. See Sanchez, 98 N.Y.2d at 380, 748 N.Y.S.2d 312, 777 N.E.2d 204 ("Register requires a significantly heightened recklessness, distinguishing it from manslaughter ...."); see also Payne, 3 N.Y.3d at 271, 786 N.Y.S.2d 116, 819 N.E.2d 634 ("[T]he reckless conduct must be `so wanton, so deficient in a moral sense of concern, so devoid of regard for the life or lives of others, and so blameworthy as to warrant the same criminal liability as that which the law imposes upon a person who intentionally causes the death of another.'" (quoting Gonzalez, 1 N.Y.3d at 469, 775 N.Y.S.2d 224, 807 N.E.2d 273)); Register, 60 N.Y.2d at 276, 469 N.Y.S.2d 599, 457 N.E.2d 704 (holding that "the depraved mind murder statute requires in addition [to reckless manslaughter] not only that the conduct which results in death present a grave risk of death but that it also occur `[u]nder circumstances evincing a depraved indifference to human life'" and construing the latter as an "additional requirement" (second alteration in original) (quoting N.Y. Penal Law § 125.25(2))).
"[I]t is not the province of a federal habeas court to reexamine state-court determinations on state-law questions." Estelle v. McGuire, 502 U.S. 62, 67-68, 112 S.Ct. 475, 116 L.Ed.2d 385 (1991). "[S]tate courts are the ultimate expositors of state law." Mullaney v. Wilbur, 421 U.S. 684, 691, 95 S.Ct. 1881, 44 L.Ed.2d 508 (1975). As we are bound by the New York Court of Appeals' construction of state law at the time petitioners' convictions became final, we conclude that the depraved indifference murder and reckless manslaughter statutes define distinct offenses.
*200 Even if Mannix and Archer were right that they could have properly been convicted under either statute, their vagueness claims would still fail. As the district courts correctly recognized, no clearly established constitutional prohibition of statutory vagueness is violated when two statutes proscribe the same conduct and a defendant is charged under the one subjecting him to greater punishment. Quite to the contrary, the Supreme Court has held that even if two statutes overlap and have "identical standards of proof," this, in and of itself, would not render them void for vagueness. Batchelder, 442 U.S. at 124, 99 S.Ct. 2198. Rather, "when an act violates more than one criminal statute, the [g]overnment may prosecute under either so long as it does not discriminate against any class of defendants," an equal protection, not due process, concern. Id. at 123-24, 99 S.Ct. 2198. Petitioners raise no such discrimination claim here. Thus, we conclude that their vagueness arguments are foreclosed by Batchelder's clear holding that "a defendant has no constitutional right to elect which of two applicable... statutes shall be the basis of his indictment and prosecution." Id. at 125, 99 S.Ct. 2198; see also United States v. Carmel, 548 F.3d 571, 579 (7th Cir.2008) (stating that "Congress may lawfully punish the same action under two separate statutes without running afoul of the Due Process Clause"); United States v. Malik, 385 F.3d 758, 760 (7th Cir.2004) (concluding that "[w]hen the same acts violate multiple laws, the prosecutor is free to choose the one with the highest sentence").[13]
Petitioners nevertheless submit that when the New York Court of Appeals overruled Register, and declared "depraved indifference" to be a culpable mental state in Feingold, the court conceded that the interpretation of the depraved indifference murder statute was unconstitutionally vague at the time their convictions were finalized. As applied to these habeas petitioners, the change in the interpretation of the statute warrants no such conclusion. Indeed, this change which does not apply retroactively, see Henry, 578 F.3d at 139-41neither diminished the notice given to petitioners that their conduct was proscribed by New York law nor authorized arbitrary enforcement of the depraved indifference statute, see generally Farrell, 449 F.3d at 484-85. Moreover, although we are of the view that the meaning of New York's depraved indifference murder statute was clear at the relevant time, it is also true that "some ambiguity in a statute's meaning is constitutionally tolerable." United States v. Chestaro, 197 F.3d 600, 605 (2d Cir.1999); accord Grayned, 408 U.S. at 110, 92 S.Ct. 2294 (declining to require "mathematical certainty" or "meticulous specificity" for statutes to survive vagueness review). Even if we were to consider petitioners' conduct under the law as it now stands, a rational jury could reasonably have found each man guilty of depraved indifference murder. Shooting into an enclosed spacewhether a small bathroom or a carknowing that people are inside, is conduct that evinces a mental state of depraved indifference to human *201 life. Feingold, 7 N.Y.3d at 294, 819 N.Y.S.2d 691, 852 N.E.2d 1163; see also Register, 60 N.Y.2d at 282, 469 N.Y.S.2d 599, 457 N.E.2d 704 (Jasen, J., dissenting).
Accordingly, we hold that the state courts did not unreasonably apply federal law in concluding (1) that, at the time their convictions became final, petitioners Mannix and Archer were on notice that their shootings were proscribed by law; and (2) that the depraved indifference murder statute did not encourage arbitrary enforcement. Petitioners have failed to identify any principle of clearly established federal law suggesting that New York's depraved indifference murder statute was void for vagueness as applied to their cases. To the contrary, Batchelder compels the conclusion that the state's decision to charge petitioners for depraved indifference murder, rather than only reckless manslaughter, does not warrant habeas relief.

C. Legal Sufficiency of the Evidence Supporting Petitioner Mannix's Conviction
Mannix's challenge to the legal sufficiency of the evidence supporting his conviction is plainly without merit.[14] Mannix cannot demonstrate that "no rational trier of fact could have found proof of guilt beyond a reasonable doubt." Einaugler v. Supreme Court of State of N.Y., 109 F.3d 836, 839 (2d Cir.1997) (internal quotation marks omitted). Nor can Mannix rebut by clear and convincing evidenceour presumption that the state court's determination of this factual issue was correct. See Leslie v. Artuz, 230 F.3d 25, 31 (2d Cir. 2000).
At trial, the people's theory was that Mannix was angry at Torruella because he "sucker-punched" him in the face and that, bent on retaliation, he recklessly fired the fatal shot through the bathroom door. As the appellate division found, "[t]he evidence warranted the conclusion that [petitioner] knowingly and deliberately fired a pistol through a door into a small, enclosed space containing the victim and a bystander." Mannix, 756 N.Y.S.2d at 34. Mannix bases his sufficiency challenge, in part, on the fact that "no one saw the gun discharge" and no one saw him "point the gun at the door or fire it." Mannix Br. at 55. However, shortly after he shot Torruella and left the bar, Mannix called the bar, inquired if he had "hit anyone," and when he learned that he had, replied: "good." Mannix, 756 N.Y.S.2d at 34.
We agree with the district court that "the evidence was easily sufficient for a jury to conclude that Mannix fired the shot into the ladies' room recklessly and not accidentally." Mannix, 390 F.Supp.2d at 294. Indeed, evidence adduced at trial revealed that the bullet that killed Torruella was fired through the center of the door at chest level. Accordingly, we find that legally sufficient evidence supports Mannix's conviction. Mannix is not entitled to habeas relief on this basis.

IV. CONCLUSION
We have considered all of petitioners' arguments on appeal, and find them to be without merit. Accordingly, the judgments of the United States District Courts for the Southern and Eastern Districts of *202 New York, denying the petitions for writs of habeas corpus, 28 U.S.C. § 2254, are hereby affirmed.
NOTES
[1]  To the extent the district court reached a contrary result based on a report and recommendation from Magistrate Judge Gabriel W. Gorenstein to which it appeared Mannix did not object, Mannix here invokes the prison mailbox rule to argue that he satisfied his objection obligation even if the objections were not received by the district court.
[2]  The judge provided the jury with the following definition of recklessness:

A person [acts] recklessly with respect to another person's death, when that person engages in conduct which creates a substantial, unjustifiable and grave risk that another person's death will occur, and when he is aware of and consciously disregards that risk.
And when that risk is of such nature and degree that disregard of it constitutes a gross deviation from the standard of conduct that a reasonable person would observe in the situation.
Trial Tr. 1068:16-25; see also N.Y. Penal Law § 15.05(3).
[3]  Petitioner Mannix was sentenced to a term of imprisonment of eighteen years for the depraved indifference murder charge on which he was convicted.
[4]  Mannix raised a number of other issues before the appellate division and before the district court. We do not discuss these issues here; we address only the issues on which our Court granted a certificate of appealability.
[5]  Petitioner Archer was sentenced to an indeterminate term of imprisonment of twenty-two years to life for the depraved indifference murder charge.
[6]  A complicated procedural history ensued between the time of Archer's conviction and the affirmance by the appellate division. These proceedings are summarized in the opinion of the district court. Archer, 2009 WL 1011591, at *4-5. Because this procedural history does not bear on our decision, there is no need to recite it here.
[7]  New York Criminal Procedure Law § 440.10(1) provides:

At any time after the entry of a judgment, the court in which it was entered may, upon motion of the defendant, vacate such judgment upon the ground that: (h) The judgment was obtained in violation of a right of the defendant under the constitution of [New York] [S]tate or of the United States.
N.Y. Crim. Proc. Law § 440.10(1)(h).
[8]  Petitioner also moved in the appellate division for a writ of error coram nobis, which was denied. People v. Archer, 19 A.D.3d 431, 795 N.Y.S.2d 899 (2d Dep't 2005). He applied for leave to appeal this denial to the New York Court of Appeals, which was also denied. People v. Archer, 5 N.Y.3d 803, 803 N.Y.S.2d 32, 836 N.E.2d 1155 (2005).
[9]  As is the case with Mannix, Archer raised numerous other issues before the state and district courts. We limit our discussion to the issue on which we granted a certificate of appealability.
[10]  Before a federal court can consider a petition for a writ of habeas corpus, the petitioner must have exhausted all available state judicial remedies. 28 U.S.C. § 2254(b)(1)(A). Exhaustion requires that a petitioner "fairly present federal claims to the state courts in order to give the State the opportunity to pass upon and correct alleged violations of its prisoners' federal rights." Duncan v. Henry, 513 U.S. 364, 365, 115 S.Ct. 887, 130 L.Ed.2d 865 (1995) (internal quotation marks and brackets omitted). "[I]t is not sufficient merely that the federal habeas applicant has been through the state courts." Picard v. Connor, 404 U.S. 270, 275-76, 92 S.Ct. 509, 30 L.Ed.2d 438 (1971). A claim is regarded as "procedurally defaulted for the purposes of federal habeas review where the petitioner failed to exhaust state remedies and the court to which the petitioner would be required to present his claims in order to meet the exhaustion requirement would now find the claims procedurally barred." Reyes v. Keane, 118 F.3d 136, 140 (2d Cir.1997) (internal quotation marks and emphasis omitted).
[11]  Moreover, the state courts' factual findings are presumed correct unless petitioners rebut the presumption with clear and convincing evidence. 28 U.S.C. § 2254(e)(1).
[12]  For purposes of AEDPA, a claim is deemed to have been "adjudicated on the merits," 28 U.S.C. § 2254(d), if the state court rendered "a decision finally resolving the [petitioner's] claim[ ], with res judicata effect, that is based on the substance of the claim advanced, rather than on a procedural, or other, ground." Sellan v. Kuhlman, 261 F.3d 303, 311 (2d Cir.2001). A state court decision will be deemed an adjudication on the merits as long as "there is nothing in its decision to indicate that the claims were decided on anything but substantive grounds." Aparicio v. Artuz, 269 F.3d 78, 94 (2d Cir.2001).
[13]  The Supreme Court has never suggested that the classic example of overlapping crimesgreater and lesser included offensesraises any due process vagueness concerns. Rather, it is the Double Jeopardy Clause that protects a defendant from conviction of both such crimes. See Brown v. Ohio, 432 U.S. 161, 168-69, 97 S.Ct. 2221, 53 L.Ed.2d 187 (1977) (treating greater and lesser offenses as the same for purposes of double jeopardy); Aparicio v. Artuz, 269 F.3d 78, 96 (2d Cir.2001) (holding that double jeopardy proscribes multiple prosecutions not only for "same offense" but "when one offense is a lesser included offense of the other").
[14]  As we have noted on countless occasions, "we must consider the evidence in the light most favorable to the prosecution and make all inferences in its favor. Moreover, petitioner bears a very heavy burden in convincing a federal habeas court to grant a petition on the grounds of insufficiency of the evidence." Fama v. Comm'r of Corr. Servs., 235 F.3d 804, 811 (2d Cir.2000)(citation omitted).
