
251 P.3d 363 (2011)
Byron R. LEU, Appellant,
v.
STATE of Alaska, Appellee.
No. A-10346.
Court of Appeals of Alaska.
April 8, 2011.
*364 Alexandra Foote-Jones, Assistant Public Defender, and Quinlan Steiner, Public Defender, Anchorage, for the Appellant.
Ann B. Black, Assistant Attorney General, Office of Special Prosecutions and Appeals, Anchorage, and Daniel S. Sullivan, Attorney General, Juneau, for the Appellee.
Before: COATS, Chief Judge, and MANNHEIMER and BOLGER, Judges.

OPINION
COATS, Chief Judge.
Byron R. Leu was convicted of fourth-degree domestic violence assault and sixth-degree misconduct involving a controlled substance.[1] He argues that his assault was justified in defense of his daughter, and that the district court erred by refusing to instruct the jury on defense of a third person. He also argues that the domestic violence statutes are unconstitutionally vague on their face and as applied to him. For the reasons discussed below, we reject Leu's claims and affirm his convictions.

Facts and proceedings
On May 23, 2008, Byron Leu was visiting his friend Kenneth Wehmeier. Leu had his eight-month-old daughter with him. The two men were drinking and "hanging out" as they often did. According to Wehmeier, he and Leu had previously been sexually involved, but that involvement had ended some five months earlier, when Leu met his girlfriend.
At some point during this visit Wehmeier made remarks that angered Leu. Wehmeier testified that Leu chased him into the kitchen and pushed him into some cabinets, causing the shelves and their contents to spill to the floor. Leu then held Wehmeier around the neck and punched him in the face several times.
Wehmeier testified that after this assault he called 911 and Leu called his girlfriend to come pick him up. During the three or four minutes it took the police to arrive, Wehmeier tried to get Leu to leave, kicking him in the legs in an effort to get him out the door. Leu refused to leave, telling Wehmeier he was waiting for his girlfriend and that he did not want to wait outside with the baby.
When the police arrived, Leu denied assaulting Wehmeier. Initially, he told the police he did not know how Wehmeier was injured, but later he said Wehmeier must have fallen down. The two officers observed that Wehmeier's house was generally tidy but that in the kitchen several pantry shelves and their contents had been upended. The officers also observed that Leu was calm and that Wehmeier was upset and crying. Wehmeier's left eye was swollen and he had a cut on his shin. The police arrested Leu, and during a pat-down search they found a small amount of marijuana, a marijuana pipe, and a digital scale in his coat pocket.
Leu was charged with fourth-degree assault and sixth-degree misconduct involving a controlled substance. At trial on those charges, Leu denied assaulting Wehmeier. He said Wehmeier got very drunk and said derogatory things about women and called his daughter names. He said he told Wehmeier he no longer wanted to be friends and that Wehmeier became irate and told him to leave. Leu said that Wehmeier then ran around the house gathering things that belonged to Leu and his daughter and throwing them out the door, including the child's car seat. Leu testified that he retrieved the car seat but when he tried to re-enter the house to get his daughter Wehmeier kicked him and tried to shove him out the door. Leu said he put his left arm around Wehmeier's neck and used his foot to trip him and that they both fell to the floor. Leu speculated that Wehmeier may have received his injuries the black eye and the cut on his shin during this fall.
*365 After the close of evidence, Leu asked the court to instruct the jury on defense of a third person (his daughter) and self-defense. District Court Judge John Wolfe refused to give the instruction on defense of a third person but agreed to instruct the jury on self-defense. The jury convicted Leu of both charges.
At sentencing, the court found by a preponderance of the evidence that Leu and Wehmeier had previously engaged in a sexual relationship, and that the assault was therefore a domestic violence assault.
Leu appeals.

Why we conclude that the district court did not err in instructing the jury
At trial Leu testified that the only reason he used force against Wehmeier was to protect his eight-month-old daughter, and he asked Judge Wolfe to instruct the jury on the justification of defense of a third person. Under AS 11.81.340, a person is justified in using force to defend a third person if he reasonably believes the third person would be justified in using that degree of force in self-defense.[2]
Leu argues that Judge Wolfe based his refusal to give the defense of a third person instruction on the legally irrelevant consideration that Leu's daughter, at eight months old, was incapable of acting in her own defense. Although we acknowledge some ambiguity in Judge Wolfe's remarks, we do not agree with Leu's interpretation of them. Judge Wolfe ultimately focused on the appropriate questionwhether Leu's child "would have been justified in throwing Mr. Wehmeier down to defend herself"[3]not on whether the child was physically capable of doing so.
To be entitled to an instruction on defense of a third person, Leu had to present "some evidence" that he reasonably believed that his daughter faced imminent harm or threat of harm.[4] In district court, Leu did not offer any evidence to support a reasonable belief that an attack on his daughter was imminent, such that the child would have been justified in using force to defend herself. On appeal, Leu argues that Wehmeier's angry and irrational behavior made it reasonable for him to conclude that Wehmeier was capable of harming his daughter. But even accepting this assertion as true, it does not amount to evidence that Leu reasonably feared that Wehmeier posed an imminent risk to his daughter.[5] Judge Wolfe therefore did not err in refusing Leu's proposed instruction on defense of a third person.
Judge Wolfe concluded that Leu did present sufficient evidence to raise a claim of self-defense because there was evidence that Wehmeier used unlawful force (kicking and shoving) to keep Leu out of the house. Judge Wolfe reasoned that Leu had a right to reenter Wehmeier's house to retrieve his daughter, and that when Wehmeier used force to keep Leu out of the house, this was arguably an unlawful use of force against Leu, thus entitling Leu to use defensive force against Wehmeier.
Leu challenges Judge Wolfe's ruling on two grounds. First, he argues that the self-defense instruction did not encompass his theory of the case and that Judge Wolfe erroneously concluded that it did. Second, he argues that the court's ruling precluded him from arguing to the jury that he acted to protect his daughter.
Leu's second claim is not supported by the record. Although Leu was not entitled to a defense of others instruction, nothing in the judge's remarks precluded Leu from arguing to the jury that he used force against Wehmeier to protect his daughter, and Leu in fact made that argument.
*366 The problem with Leu's other claimthat Judge Wolfe erred in concluding that the self-defense instruction encompassed his theory of the caseis that Leu failed to object on this ground in district court.[6]
Leu is not challenging Judge Wolfe's decision to instruct the jury on self-defensehe requested the self-defense instruction. His quarrel is with Judge Wolfe's conclusion that the instruction adequately conveyed his theory that the force he used against Wehmeier was justified to protect his daughter.[7] But absent any objection by Leu, or any proposal to modify the language of the instruction, Judge Wolfe could reasonably conclude that Leu was satisfied that the self-defense instruction would serve this purpose.
Moreover, Leu has failed to show how he was prejudiced. In closing argument, Leu abandoned any claim that his assault on Wehmeier was justified, arguing instead that the State had failed to prove the mens rea element of fourth-degree assault. Leu argued that, because he acted reasonably when he knocked Wehmeier down to get to his daughter, the State failed to prove that he acted with the required reckless intent. The State argued that the assault occurred in the kitchen, and was entirely unrelated to Leu's effort to re-enter the house to retrieve his daughter. Given this record, we conclude that any deficiency in the court's self-defense instruction did not contribute to Leu's conviction.

Why we conclude that the definition of "household member" is not unconstitutionally vague
At sentencing, at the State's request and over Leu's objection, Judge Wolfe found by a preponderance of the evidence that Leu's assault on Wehmeier qualified as a domestic violence crime as defined in AS 18.66.990(3).[8]
The Alaska Statutes define certain crimes (e.g., assault, burglary, criminal mischief) as domestic violence crimes if they are committed by one household member against another.[9] The term "household member" is commonly understood to refer to individuals who live together. But the Alaska Statutes define the term much more broadly to include[10]:
(A) adults or minors who are current or former spouses;
(B) adults or minors who live together or who have lived together;
(C) adults or minors who are dating or who have dated;
(D) adults or minors who are engaged in or who have engaged in a sexual relationship;
(E) adults or minors who are related to each other up to the fourth degree of consanguinity, whether of the whole or half blood or by adoption, computed under the rules of civil law;
(F) adults or minors who are related or formerly related by marriage;
(G) persons who have a child of the relationship; and
(H) minor children of a person in a relationship that is described in (A)(G) of this paragraph[.]
Leu challenges subsection (D), which defines household member to include "adults *367 who are engaged in or who have engaged in a sexual relationship." He argues that this definition is unconstitutionally vague on its face and as applied to him.
When no first amendment rights are implicated, we consider two factors in determining whether a statute is unconstitutionally vague: first, whether the statute gives adequate notice of the conduct that is prohibited and, second, whether the statute's language is so imprecise that it encourages arbitrary enforcement by allowing prosecuting authorities too much discretion in determining the scope of the law.[11] Leu attacks AS 18.66.990(5)(D) on both of these grounds.[12]
Basic fairness requires that statutes provide ordinary citizens with adequate notice of what conduct is prohibited.[13]
But the fact that people can, in good faith, litigate the meaning of a statute does not necessarily (or even usually) mean that the statute is so indefinite as to be unconstitutional. The question is whether the statute's meaning is unresolvably confused or ambiguous after it has been subjected to legal analysis.[14]
A statute will not be invalidated on its face on vagueness grounds unless it is impermissibly vague in all its applications.[15] In other words, "the possibility of difficult or borderline cases will not invalidate a statute where there is a hard core of cases to which the ordinary person would doubtlessly know the statute unquestionably applies."[16]
The term "sexual relationship" is not so unresolvably confused or ambiguous that it gives inadequate notice in all its applications of what conduct it encompasses. An ordinary person would, at a minimum, understand the term "sexual relationship" to include individuals who are in a committed sexual relationship.[17] We therefore reject Leu's claim that the term is impermissibly vague on its face.[18]
Leu also argues that the term is unconstitutionally vague as applied to him because it did not provide him with adequate notice that the term would include the type of casual sexual relationship he had with Wehmeier.
As we already explained, the void for vagueness doctrine requires that criminal laws give ordinary citizens fair notice of what conduct is prohibited, so that they are not left to guess at whether a certain course of conduct is criminal.[19] But the doctrine only requires fair notice of the elements that make the conduct criminala defendant has no right to notice of the collateral circumstance that his offense might be classified as a domestic violence crime.
We addressed a related principle in Bell v. State,[20] a case in which the defendant was convicted of inducing a girl under the age of sixteen to engage in prostitution. Bell argued that his conviction violated due process because the trial court did not allow him to *368 raise the defense of mistake of age.[21] In addressing that claim, we acknowledged the well-recognized rule that a defendant cannot be convicted of criminal conduct without an awareness or consciousness of wrongdoing.[22] But we explained that "[w]hat is essential is not awareness that a given conduct is a `wrongdoing' in the sense that it is proscribed by law, but rather, an awareness that one is committing the specific acts which are defined by law as a `wrongdoing.'"[23] Thus, for Bell's conviction to be constitutional, the State had to prove that Bell was aware he committed an act the law defined as wrongdoing: procuring a woman to engage in prostitution.[24] The State did not have to prove that Bell was aware the girl he procured was under sixteen, even though that circumstance aggravated his offense.[25] In Ortberg v. State,[26] we similarly held that there was no due process violation in convicting the defendant of second-degree criminal mischief without proof that the defendant knew or had reason to believe the damage he caused was in excess of the $500 specified in the statute.[27]
The same principle leads us to reject Leu's claim that his due process rights were violated because he was convicted of domestic violence assault without adequate notice that his offense was a crime of domestic violence. All the constitution required was proof that Leu was aware he committed an act the law defined as wrongdoing: assaulting Wehmeier. It was not necessary for Leu to be aware that, because of his prior sexual relationship with Wehmeier, the assault would be a domestic violence crime.
This does not mean Leu would have no recourse if the record left us uncertain that the legislature intended the term "sexual relationship" to apply under the facts of his case, such that we doubted whether trial judges could apply the term in a reasonably evenhanded manner. But the record does not leave us with that doubt.
At trial, Leu denied that he and Wehmeier had ever been sexually involved. But Judge Wolfe credited Wehmeier's testimony about the nature of the two men's relationship, and Leu does not challenge that finding on appeal. Wehmeier testified that during the year or so he lived next door to Leu they "used to hang out together a lot." He said they were "both drinkers" and would "stay up pretty late, you know, watching TV or whatever, and we would end up in the same bed." Wehmeier said that "pretty soon we were intimate, but we were more like friends. I mean, that was something that would happen every once and a while." Wehmeier said he and Leu had not been "intimate" for about five months before the incident in this case, when Leu met his girlfriend. Wehmeier said he was jealous at first and that it took him a few months to warm up to Leu's girlfriend. But he testified that "the friendship . . . was what was important to me." On cross-examination, Wehmeier acknowledged that he had called Leu and been to his house a few times since the assault; he said he "thought we maybe could patch things up," and that he "wanted to forgive [Leu]."
Based on this record, Leu asserts that any sexual involvement he had with Wehmeier was too casual to fall within the definition of "sexual relationship." But when the Alaska Legislature defined "household member" to include adults "who have engaged in a sexual relationship," it did not specify that the relationship had to be a serious one. As Leu points out, Minnesota has imposed this requirement by statute, and it has adopted a multi-factor test to determine when a romantic or sexual relationship is "significant."[28] The Alaska Legislature chose not to define *369 "sexual relationship," leaving the courts to construe the term in accordance with its ordinary meaning.[29]
As just explained, Judge Wolfe credited Wehmeier's testimony that the two men had an ongoing friendship that, up until five months before the assault, included occasional sexual intimacy, and that this intimacy continued until Leu met his girlfriend. This is not the type of non-consensual or short-lived sexual involvement that falls outside the ordinary person's understanding of a "sexual relationship."
Leu's next argument is that the term "sexual relationship" is impermissibly vague because it lends itself to arbitrary or selective enforcement. Because Leu did not raise this claim in the district court, he must show plain error.[30]
Leu does not argue that the police or prosecuting authorities have actually engaged in arbitrary or selective enforcement in their prosecution of domestic violence crimes; nor does he provide any evidence of such conduct. The Alaska Supreme Court has held that there must be actual evidence of a history of arbitrary or capricious enforcement to invalidate a statute on this ground, or the language of the statute must be so conflicting and confused that arbitrary enforcement is inevitable.[31] A law will not be invalidated simply because of its potential for arbitrary enforcement.[32]
Leu tries to get around this requirement by arguing that the cases in which a history of selective and arbitrary enforcement has been established usually involve "disorderly conduct type laws" that historically have been used to "target undesirables." Leu argues that he should be exempt from this requirement because this historical context "is simply not present in the context of domestic violence cases." But we have never held that a defendant is required to offer evidence of selective or arbitrary enforcement only if the statute at issue has historically been used to "target undesirables."[33] Furthermore, the term "sexual relationship" "is not so imprecise that it obviously would encourage arbitrary enforcement."[34] We find no plain error.

Conclusion
We AFFIRM Leu's convictions.
NOTES
[1]  AS 11.41.230(a)(1) and AS 11.71.060(a)(1), respectively.
[2]  AS 11.81.340 provides:

A person is justified in using force upon another when and to the extent the person reasonably believes it is necessary to defend a third person when, under the circumstances as the person claiming defense of another reasonably believes them to be, the third person would be justified under AS 11.81.330 or 11.81.335 in using that degree of force for self-defense.
[3]  Emphasis added.
[4]  AS 11.81.340; AS 11.81.900(b)(27); Ha v. State, 892 P.2d 184, 191 (Alaska App. 1995).
[5]  See Lamont v. State, 934 P.2d 774, 777-79 (Alaska App. 1997) (citing McCracken v. State, 914 P.2d 893, 898 (Alaska App. 1996)).
[6]  See Alaska R.Crim. P. 30(a) ("No party may assign as error any portion of the charge or omission therefrom unless the party objects thereto before the jury retires to consider its verdict.").
[7]  See Carman v. State, 658 P.2d 131, 135 (Alaska App.1983).
[8]  See Alaska Criminal Rule 32(e), which provides:

Judgment for Crimes Involving Domestic Violence. In a case in which the defendant is convicted of an offense listed in AS 18.66.990(3) and the prosecution claims at sentencing that the offense is a crime involving domestic violence, the written judgment must set forth whether the offense is a crime involving domestic violence as defined in AS 18.66.990(3) and (5). A factual and legal determination supporting this finding must be made on the record.
[9]  AS 18.66.990(3).
[10]  AS 18.66.990(5); see Cooper v. District Court, 133 P.3d 692, 707 (Alaska App.2006); Bingaman v. State, 76 P.3d 398, 406-08 (Alaska App.2003); Carpentino v. State, 42 P.3d 1137, 1141 (Alaska App.2002).
[11]  Summers v. Anchorage, 589 P.2d 863, 866-67 (Alaska 1979).
[12]  To the extent that Leu is raising a separate claim of overbreadth, that claim is waived for inadequate briefing. See Petersen v. Mutual Life Ins. Co. of N.Y., 803 P.2d 406, 410 (Alaska 1990) ("Where a point is not given more than a cursory statement in the argument portion of a brief, the point will not be considered on appeal."); see also Holton v. State, 602 P.2d 1228, 1236 n. 11 (Alaska 1979) & Marks v. Anchorage, 500 P.2d 644, 646 (Alaska 1972) (recognizing that claims of vagueness and overbreadth are "functionally and doctrinally distinct").
[13]  State v. Rice, 626 P.2d 104, 109 (Alaska 1981); Stock v. State, 526 P.2d 3, 8 (Alaska 1974).
[14]  De Nardo v. State, 819 P.2d 903, 907 (Alaska App. 1991) (emphasis in original).
[15]  Village of Hoffman Estates v. Flipside, Hoffman Estates, Inc., 455 U.S. 489, 494-95, 102 S.Ct. 1186, 1191, 71 L.Ed.2d 362 (1982); Bachlet v. State, 941 P.2d 200, 204 (Alaska App.1997).
[16]  Stock, 526 P.2d at 9.
[17]  See Webster's New World College Dictionary (4th ed. 2007) (defining "relationship" as "a continuing attachment or association between persons. . ., specif., one between lovers").
[18]  See Stock, 526 P.2d at 9.
[19]  Gottschalk v. State, 575 P.2d 289, 290 (Alaska 1978).
[20]  668 P.2d 829 (Alaska App. 1983).
[21]  Id. at 832-35.
[22]  Id. at 833.
[23]  Id.
[24]  Id.
[25]  Id.
[26]  751 P.2d 1368 (Alaska App. 1988).
[27]  Id. at 1374.
[28]  Minn.Stat. § 518B.01(b) (defining "family or household members" to include "persons involved in a significant romantic or sexual relationship" and adopting factors for the court to consider in determining if the relationship is significant).
[29]  See Anchorage v. Suzuki, 41 P.3d 147, 150 (Alaska 2002) (discussing rules of statutory construction); Norman J. Singer and J.D. Shambie Singer, 2A Statutes and Statutory Construction, § 47:7 at 306-11 (7th ed. 2007).
[30]  See Alaska R.Crim. P. 47(b); Garroutte v. State, 508 P.2d 1190, 1191 (Alaska 1973).
[31]  Lazy Mountain Land Club v. Matanuska-Susitna Borough Bd. of Adjustment and Appeals, 904 P.2d 373, 384 (Alaska 1995) (citing Summers, 589 P.2d at 869 and Williams v. State, Dep't of Revenue, 895 P.2d 99, 105 (Alaska 1995)).
[32]  Haggblom v. City of Dillingham, 191 P.3d 991, 998 (Alaska 2008).
[33]  See, e.g., Bachlet, 941 P.2d at 206 (receipt of bribe statute); Morrow v. State, 704 P.2d 226, 233 (Alaska App. 1985) (imitation of controlled substance statute); McKenzie v. Anchorage, 631 P.2d 514, 518 (Alaska App. 1981) (gambling ordinances).
[34]  See Turney v. State, 936 P.2d 533, 544 (Alaska 1997) (quoting Summers, 589 P.2d at 867).
