Kody Brown is a proud polygamist, and a relatively famous one. Now Mr. Brown, his four wives and 16 children and stepchildren are going to court to keep from being punished for it.

The family is the focus of a reality TV show, “Sister Wives,” that first appeared in 2010. Law enforcement officials in the Browns’ home state, Utah, announced soon after the show began that the family was under investigation for violating the state law prohibiting polygamy.

On Wednesday, the Browns are expected to file a lawsuit to challenge the polygamy law.

The lawsuit is not demanding that states recognize polygamous marriage. Instead, the lawsuit builds on a 2003 United States Supreme Court decision, Lawrence v. Texas, which struck down state sodomy laws as unconstitutional intrusions on the “intimate conduct” of consenting adults. It will ask the federal courts to tell states that they cannot punish polygamists for their own “intimate conduct” so long as they are not breaking other laws, like those regarding child abuse, incest or seeking multiple marriage licenses.

Mr. Brown has a civil marriage with only one of his wives; the rest are “sister wives,” not formally wedded. The Browns are members of the Apostolic United Brethren Church, a fundamentalist offshoot of the Mormon Church, which gave up polygamy around 1890 as Utah was seeking statehood.