Salih Sevencan appeals from an Opinion and Order of the District Court for the Eastern District of New York (Allyne R. Ross, Judge) entered on August 3, 2001, denying his petition for a writ of habeas corpus. The District Court granted a Certificate of Appealability on the issue of whether the trial court’s refusal to except Sevencan’s wife from a limited courtroom closure order violated Sevencan’s Sixth Amendment rights.
We hold that (1) the District Court properly conducted a Nieblas hearing in order to determine that the exclusion of Seven-can’s wife was justified and (2) the state trial court’s decision to exclude Sevencan’s wife was not “an unreasonable application of[] clearly established Federal law, as determined by the Supreme Court of the United States,” 28 U.S.C. § 2254(d).
I. BACKGROUND
We assume familiarity with the background facts of this case as set forth in the District Court’s opinion, Sevencan v. Herbert, 152 F.Supp.2d 252 (E.D.N.Y.2001). We briefly set forth here only those facts necessary to our disposition.
Between June and September 1993 Sev-encan and four co-defendants were tried in the Supreme Court for Kings County, New York, on various charges stemming from their participation in a conspiracy to import heroin from Turkey and sell it in the United States. Among the crimes that the defendants were charged with were weapons possession and conspiracy to commit murder.
The chief witness against Sevencan was an undercover police officer. Before the officer testified, the prosecution moved to seal the courtroom during the officer’s testimony. Defense counsel for Sevencan objected. The trial court held a hearing, and, on June 16, 1993, it granted the motion to seal the courtroom during the officer’s testimony. In so ruling, the trial judge noted the importance of an open courtroom, but found the need to protect the identity of the undercover officer for his safety compelling, in light of the officer’s continuing undercover work. The judge made an exception for attorneys and law student interns working with defense counsel, stating:
a lawyer is a lawyer, and we’[re] sorry about spectators, family, I mean, another matter, something that [the prosecutor] can’t exclude from the courtroom is the two defendants on bail, they’re going to walk out, I mean, the family — there’s always a certain amount of risk in everything we do and we try to do it.
*72152 F.Supp.2d at 255 (alterations in original).
On June 24, 1993, the court informed counsel that both the undercover officer and the assistant district attorney had received death threats connected to the trial. Twelve days later, Sevencan’s wife attempted to attend the trial. Defense counsel sought an exception to the sealing order at sidebar, which the Court denied as follows:
[Defense Counsel]: Judge, the defendant’s wife is here and works all the time. This is practically the only day she can get here and would like to come in. I understand-
The Court: The courtroom is sealed. [Defense Counsel]: Yes, it is. That’s why I’m applying to you [so that] it be allowed-
The Court: The reason we seal it is to protect the undercover. I don’t think she falls within the exceptions I’ve created. What’s the district attorney’s position?
[The Prosecutor]: I would object.
The Court: Your application is denied.
Id. at 255-56.
After Sevencan was convicted, he argued on appeal, inter alia, that the closure of the courtroom, including the exclusion of his wife, deprived him of this Sixth Amendment rights. The Appellate Division of the Supreme Court rejected his arguments as follows:
The defendant contends that he was denied his right to a public trial when the trial court closed the courtroom during the testimony of two undercover police officers. However, his present claim was waived by his failure to object to the closures at trial, and, in any event, is without merit.
People v. Sevencan, 258 A.D.2d 485, 685 N.Y.S.2d 735, 736 (2d Dep’t 1999) (citing People v. Akaydin, 258 A.D.2d 466, 685 N.Y.S.2d 737 (2d Dep’t 1999) (companion case)). Notably, although the Appellate Division appeared to hold that Sevencan had not objected to the closures at trial, it did not similarly hold that his co-defendant, Akaydin, had also waived his objection to those closures. See Akaydin, 685 N.Y.S.2d at 738. Yet it appears from the record that counsel for Sevencan objected more vocally than counsel for Akaydin. Sevencan, 152 F.Supp.2d at 260. See generally N.Y.Crim. Proc. Law § 470.05(2).1 The New York Court of Appeals denied leave to appeal further. See People v. Sevencan, 93 N.Y.2d 1027, 697 N.Y.S.2d 586, 719 N.E.2d 947 (1999).
Sevencan subsequently timely filed the instant petition in the District Court. He sought relief on various grounds, including the closure of the courtroom and the exclusion of his wife. The proceedings subsequently focused on the exclusion of Seven-can’s wife.
Inasmuch as the state trial court had made no findings specific to Sevencan’s wife, the District Court held a hearing pursuant to Nieblas v. Smith, 204 F.3d 29 (2d Cir.1999) (holding that district courts *73have the discretion to conduct evidentiary hearings to determine whether a courtroom closure was justified where the record of proceedings before the trial court was not sufficient to determine whether the closure was proper). Sevenean, 152 F.Supp.2d at 263-64. At the hearing, the State submitted tapes of conversations that suggest Mrs. Sevencan’s knowledge of her husband’s illegal activities and her familiarity with several of Sevencan’s associates. Id. at 258-59. The undercover officer also testified, stating that he did business at bars in a large shopping area for residents of Mrs. Sevencan’s neighborhood, that he spent between two and five days a week in this area, and that he intended to continue working in that area under cover. He further testified that the leader of the drug conspiracy was still at large at the time of trial and that Seven-can himself had previously committed a murder.
Following the Nieblas hearing, the District Court held that the closure of the courtroom to Mrs. Sevenean during one day of the undercover officer’s testimony did not deny Sevenean his right to a public trial because (1) protection of the undercover’s safety and security was an important interest within the meaning of Waller v. Georgia, 467 U.S. 39, 48, 104 S.Ct. 2210, 81 L.Ed.2d 31 (1984), and (2) the State demonstrated that Mrs. Sevenean was likely to encounter the undercover officer in the course of her daily activities. Sevencan, 152 F.Supp.2d at 264-65. Although the District Court acknowledged that Mrs. Sevenean did not pose a direct physical threat to the undercover officer, it held that her ability to report that officer’s description to other more dangerous individuals was sufficient. Id. at 265-66. The District Court also found that Mrs. Seven-can is “a timid[,] pliable woman” likely to be susceptible to requests from her husband’s associates, who still were at large, to inform them if she spotted the undercover officer. Id. at 266-67.
The District Court then rejected Seven-can’s remaining claims and denied his petition. Id. at 269-70. It granted a Certificate of Appealability limited to the issue of whether the exclusion of Sevencan’s wife deprived him of his Sixth Amendment right to a public trial. See id. at 270.
On appeal, Sevenean principally argues that (1) the District Court erred in holding a hearing pursuant to Nieblas v. Smith and, accordingly, the writ should have been granted based on the lack of findings by the state trial court; and (2) even if the evidence adduced at the Nieblas hearing was properly considered, the District Court erred in holding that the evidence and record before it were sufficient to support the state trial court’s decision to exclude Sevencan’s wife from the courtroom.
II. Discussion
The habeas corpus statute available to persons convicted under state law, as amended by the Antiterrorism and Effective Death Penalty Act of 1996 (“AED-PA”), Pub.L. No. 104-132, 110 Stat. 1214, provides, in relevant part:
An application for a writ of habeas corpus on behalf of a person in custody pursuant to the judgment of a State court shall not be granted with respect to any claim that was adjudicated on the merits in State court proceedings unless the adjudication of the claim-
(1) resulted in a decision that was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States.
28 U.S.C. § 2254(d) (emphasis added). In Williams v. Taylor, 529 U.S. 362, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000), the Supreme *74Court authoritatively interpreted the phrase “clearly established Federal law, as determined by the Supreme Court of the United States” to mean “the holdings, as opposed to the dicta, of [the Supreme Court’s] decisions as of the time of the relevant state-court decision.” 529 U.S. at 412, 120 S.Ct. 1495. “[A]s the statutory language makes clear, v . 2254(d)(1) restricts the source of clearly established law to [the Supreme Court’s] jurisprudence.” Id.
In Waller v. Georgia, 467 U.S. 39, 104 S.Ct. 2210, 81 L.Ed.2d 31 (1984), the Supreme Court held that, before public access to a courtroom in a criminal case may be restricted, “[1] the party seeking to close the hearing must advance an overriding interest that is likely to be prejudiced, [2] the closure must be no broader than necessary to protect that interest, [3] the trial court must consider reasonable alternatives to closing the proceeding, and [4] [the court] must make findings adequate to support the closure.” Id. at 48, 104 S.Ct. 2210. The violation of the constitutional right to a public trial is a “structural error”2 warranting remediation regardless of prejudice. See Id. at 49-50 & n. 9, 104 S.Ct. 2210.
Prior to the enactment of AEDPA, we set forth a standard in our own Circuit governing the constitutionality of excluding a defendant’s family members from the courtroom. See Vidal v. Williams, 31 F.3d 67 (2d Cir.1994). In Vidal, we reversed a District Court’s denial of a habeas petition on the ground that the closure of a courtroom to a defendant’s parents was improper. In that case, the defendant’s parents were in the courtroom at the time the prosecutor sought closure, and defense counsel requested that they be allowed to remain. Id. at 68. The state trial court had rejected that request because the parents lived within three miles of the area in which the officer conducted undercover operations, and it closed the courtroom. Id. We held that the proximity of the parents’ residence to the officer’s area of operation was insufficient to justify their exclusion, relying in part on the Supreme Court’s “specific[ ] notation, in dicta, of] a special concern for assuring the attendance of family members of the accused” in In re Oliver, 333 U.S. 257, 271-72 & n. 29, 68 S.Ct. 499, 92 L.Ed. 682 (1948). Vidal, 31 F.3d at 69.
Since Vidal, we have often noted in pre-AEDPA cases that particular attention must be paid to the exclusion of family members. See English v. Arbuz, 164 F.3d 105, 108 (2d Cir.1998) (granting a habeas petition in a pre-AEDPA case where the prosecutor sought closure of the courtroom, the defendant asked that his family be allowed to remain, and the state trial court denied the defendant’s request and closed the courtroom); Guzman v. Scully, 80 F.3d 772, 775-76 (2d Cir.1996) (similar, citing Oliver); see also Bowden v. Keane, 237 F.3d 125, 130 n. 1 (2d Cir.2001) (“Special concerns may apply when the spectators selectively barred from the courtroom are the defendant’s family members.” (citing Vidal)); Brown v. Kuhlmann, 142 F.3d 529, 538 (2d Cir.1998) (similar, citing Vidal and Oliver); Ayala v. Speckard, 131 F.3d 62, 72 (2d Cir.1997) (en banc) (similar, citing Vidal and Guzman).
In Yung v. Walker, 341 F.3d 104, 111 (2d Cir.2003) — the first and only decision of our Court on the exclusion of family members applying the AEDPA standard (as *75informed by Williams v. Taylor) — we held that “Waller prevents a court from denying a family member’s request to be exempted from a courtroom closure order unless the court is convinced that the exclusion of that particular relative is necessary to protect the overriding interest at stake.”3 We noted, however, that the particular showing required in Vidal was not mandated by “clearly established” Supreme Court law. Because the District Court had relied on Vidal, we vacated the judgment and remanded for application of “the more general teachings of Waller as informed by Oliver. ” Id.
In the case now before us, the District Court likewise followed Vidal and analyzed the issue as if “clearly established” law includes our Circuit’s pre-AEDPA jurisprudence regarding the consideration due to family members in such circumstances. See Sevenean, 152 F.Supp.2d at 262-63. This was error. As we recognized in Yung, the only “clearly established” law within the meaning of 28 U.S.C. § 2254(d)(1) relevant to this case is Waller ’s four-factor test for closures generally. See Yung, 341 F.3d at 111. Accordingly, Sevenean is entitled to relief only if he can demonstrate that the state court’s decision “resulted in a decision that was contrary to, or involved an unreasonable application of,” Waller. See 28 U.S.C. § 2254(d)(1).
A state court’s decision is “contrary to” clearly established law, within the meaning of § 2254(d)(1), if “the state court applies a rule that contradicts” governing Supreme Court law, or if it “confronts a set of facts that are materially indistinguishable from a decision of [the Supreme] Court and nevertheless arrives at a result different from [Supreme Court] precedent.” Williams, 529 U.S. at 405, 406, 120 S.Ct. 1495. Ordinarily, a “run-of-the-mill state court decision applying the correct legal rule from our cases to the facts of a prisoner’s ease would not fit comfortably within § 2254(d)(l)’s ‘contrary to’ clause,” even if a federal court believes that the state court reached the wrong result. Id. at 406, 120 S.Ct. 1495. This is just such a “run-of-the-mill” case — indeed, petitioner does not contend that the result is “contrary to” federal law, but, rather, that the state appellate court “failfed] to [properly] apply” federal law. See Pet’r’s Br. at 29. This claim is properly analyzed pursuant to the “unreasonable application” clause of 28 U.S.C. § 2254(d). See Williams v. Taylor, 529 U.S. at 407, 120 S.Ct. 1495.
A decision is an “unreasonable application” of clearly established Supreme Court law if a state court “identifies the correct governing legal principle from [the Supreme Court’s] decisions but unreasonably applies that principle to the facts of [a] prisoner’s case.” Id. at 413, 120 S.Ct. 1495. Under this standard, “a federal habeas court may not issue the writ simply because that court concludes in its independent judgment that the relevant state-court decision applied clearly established federal law erroneously or incorrectly. Rather, that application must also be unreasonable.” Id. at 411, 120 S.Ct. 1495. In order to grant the writ there must be “[s]ome increment of incorrectness beyond error,” although “the increment need not *76be great; otherwise, habeas relief would be limited to state court decisions so far off the mark as to suggest judicial incompetence.” Francis S. v. Stone, 221 F.3d 100, 111 (2d Cir.2000) (internal quotation marks omitted).
In this case, the state trial court’s initial decision to close the courtroom to the general public clearly comported with Waller: (1) the prosecutor advanced an “overriding interest” that was “likely to be prejudiced” by a public hearing — the safety of the undercover officer; (2) the closure was “no broader than necessary to protect that interest,” as it was solely for the officer’s testimony; (3) the trial court considered reasonable alternatives proposed by the defense — the admission of other lawyers and legal interns — and (4) the trial court made “findings adequate to support the closure,” in that it found that the undercover officer would indeed be in danger if he were publicly identified. See Waller, 467 U.S. at 48, 104 S.Ct. 2210. Accordingly, Sevenean is entitled to relief only if the trial court’s refusal to make an exception to its general closure order for his wife was an “unreasonable” application of Waller.
In Yung, we set forth the standard for determining whether a state court has unreasonably applied Waller by excluding a relative of the defendant from the courtroom. We held that “it would be an unreasonable interpretation of Waller for a court to deny [the] request [of a member of the defendant’s family to be exempted from a courtroom closure order] if the exclusion of that particular relative, under the specific circumstances at issue, is not necessary to promote the overriding interest [at stake].” Yung, 341 F.3d at 111.
In this ease, the state trial court did not make express findings regarding whether the exclusion of Mrs. Sevenean was necessary to protect the undercover officer’s safety. Accordingly, the District Court held a Nieblas hearing to determine whether Mrs. Sevencan’s exclusion was justified. As a preliminary matter, Seven-can argues that the District Court’s decision to hold the Nieblas hearing was in error. He advances several justifications for this assertion, each of which we consider in turn.
First, Sevenean maintains that evidence adduced at a Nieblas hearing cannot satisfy Waller because the fourth prong of the Waller test requires the state trial court itself to make “findings adequate to support the closure.” Waller, 467 U.S. at 48, 104 S.Ct. 2210. We have previously rejected this argument. See Nieblas, 204 F.3d at 31-32; see also Yung, 341 F.3d at 112; Gonzalez v. Quinones, 211 F.3d 735, 738 (2d Cir.2000).
Sevenean also argues that § 2254(d)(1) precludes district courts from conducting any ex post evidentiary hearings in habeas cases. But § 2254(d)(1) sets forth a legal standard governing when a writ can and cannot be granted: It does not explicitly or implicitly invalidate the district courts’ long-standing practice of holding evidentia-ry hearings to supplement an incomplete state record. Moreover, although Nieblas itself is a pre-AEDPA case, our opinions in both Yung and Gonzalez recognized the validity of Nieblas hearings in habeas cases filed after AEDPA’s enactment. Accordingly, Sevencan’s argument that § 2254(d)(1) precludes the use of Nieblas hearings is without merit.
Finally, Sevenean argues that even if Nieblas hearings are permissible in post-AEDPA cases generally, the facts of his case in particular did not justify a Nieblas hearing. This argument also fails: The hearing in this case was well within the scope of the District Court’s discretion as described in Nieblas. Two factors sup*77ported our affirmance in Nieblas of the district court’s decision to hold a hearing. First, the petitioner had made only a perfunctory objection at trial to the courtroom closure. Second, the law changed after trial, creating a greater necessity for a fully-developed record. We held that “where either of the above two reasons or any other similar reason exists, it is particularly appropriate for a habeas court to gather additional evidence — rather than granting the defendants the ‘windfall’ of a new trial — where the alleged constitutional violation does not affect the fairness of the outcome at trial, as in courtroom closure cases like this one.” Nieblas, 204 F.3d at 32 (quoting Waller, 467 U.S. at 50, 104 S.Ct. 2210).
Like Nieblas, Sevencan made only a perfunctory objection to the exclusion of his wife. He did not alert the trial court that a different standard might govern his wife’s exclusion. In addition, after Sevencan’s trial in 1993, the New York Court of Appeals decided People v. Nieves, 90 N.Y.2d 426, 660 N.Y.S.2d 858, 683 N.E.2d 764 (1997), in which it held that, in order to exclude a defendant’s wife and children, the People must establish “a ‘substantial probability’ that the officer’s safety would be jeopardized by [their] presence.” Nieves, 660 N.Y.S.2d at 861, 683 N.E.2d 764. As in Nieblas, it is likely that the State would have built a stronger case for the exclusion of Mrs. Sevencan had the rule of Nieves been New York law at the time of trial. Thus, the District Court did not err by holding a Nieblas hearing.
We also agree with the District Court’s conclusion, based on the evidence adduced at the Nieblas hearing, that the exclusion of Sevencan’s wife was “necessary to protect the overriding interest” in the safety of the undercover officer. Yung, 341 F.3d at 111. At the hearing, the undercover officer testified that, at the time of Sevencan’s trial, he had intended to return to the same general area in which Sevencan plied his narcotics trade. The officer stated that he had conducted undercover work at bars in a large shopping area for residents of Mrs. Sevencan’s neighborhood and that he had spent between two and five days a week in this area. Based on this evidence, the District Court properly determined that Mrs. Sev-encan was likely to have encountered the undercover officer in the course of her daily activities.
Nor do we find error in the District Court’s conclusion that Mrs. Sevencan is “a timid[,] pliable woman” likely to be susceptible to requests from her husband’s associates to inform them if she spotted the undercover officer. Sevencan, 152 F.Supp.2d at 266-67. At the Nieblas hearing, the State submitted tapes of conversations that suggest Mrs. Sevencan’s knowledge of her husband’s illegal activities and her familiarity with several of his associates. Id. at 258-59. The evidence also plainly established that Sevencan and his associates were highly dangerous: The undercover officer and the assistant district attorney both received death threats several days before Mrs. Sevencan asked to be admitted into the courtroom. There is no reason to believe that Sevencan’s associates would not make similar threats in order to force Mrs. Sevencan to identify the undercover officer. Accordingly, the evidence adduced at the Nieblas hearing overwhelmingly supports the necessity of excluding Sevencan’s wife from the courtroom in order to protect the undercover officer’s safety. In conformity with our decision in Yung, we hold that the state trial court reasonably applied the Supreme Court’s decision in Waller when it declined to exempt Mrs. Sevencan from the courtroom closure order.
*78III. CONCLUSION
In sum, we hold that (1) the District Court properly conducted a Nieblas hearing in order to determine that the exclusion of Sevencan’s wife was justified and (2) the state trial court’s decision to exclude Sevencan’s wife was not “an unreasonable application of[ ] clearly established Federal law, as determined by the Supreme Court of the United States,” 28 U.S.C. § 2254(d).
Accordingly, the judgment of the District Court is affirmed.