EBEL, Circuit Judge.
Wendy Faustin is an abortion protester. On several occasions, she has held a banner reading “ABORTION KILLS CHILDREN” on a highway overpass in Denver for motorists traveling below to see.
After Denver police repeatedly asked Faustin to stop displaying the banner, Faustin brought this action under 42 U.S.C. § 1988 alleging violations of her free speech rights under the First Amendment. In defense, Denver points to what it asserts is its unwritten, but constitutional, city policy banning signs and banners on overpasses. According to Faustin, the policy is broader and unconstitutionally bans all expression on all overpasses. Both sides originally filed motions for summary judgment below and argue summary judgment is the proper vehicle for deciding this case.
This is the second appeal in this case. Before us now are Faustin’s facial challenges to the city’s policy as overly broad and vague. The district court granted Faustin summary judgment on these claims. At this juncture, there remains some factual dispute as to the exact scope of Denver’s unwritten policy. Nevertheless, this factual dispute is not material and so will not preclude summary judgment. Even giving the policy the broadest interpretation the record will allow, Faus-tin has not shown an unconstitutional chilling effect on the protected speech of third parties not before the court.
[1195]*1195Therefore, we conclude that Denver, rather than Faustin, is entitled to judgment as a matter of law. We REVERSE and REMAND with instructions that summary judgment be entered for Defendants.
I. Background
We need not set out in detail the facts or procedural history of this case, as they were stated in our prior panel decision. See Faustin v. City and County of Denver, 268 F.3d 942, 945-47 (10th Cir.2001) (“Faustin I ”). We present only essential facts as necessary to address the issues in this appeal.
Wendy Faustin regularly engages in abortion protest activities in Denver. This litigation ensued after Denver police repeatedly prevented Faustin from displaying a large banner with the message “ABORTION KILLS CHILDREN” on a highway overpass in Denver. Faustin and a friend stood on the sidewalk between the roadway and a chain-link fence, on the overpass located at the juncture of Sixth Avenue and Perry Street, and held the banner up by hand so that motorists driving below could view it.
The parties stipulate to four specific confrontations between Faustin and Denver police. In each instance, Denver police approached Faustin and asked her to leave the overpass and/or to remove her banner. The police officers were unable to cite a specific law proscribing Faustin’s display. During the third encounter, however, a Denver police officer did inform Faustin she could not display the banner because to do so was a violation of a Denver posting ordinance. Lieutenant Donald Fink also allegedly said, over the radio to the officer at the scene, that Faustin’s display could be in violation of a state unauthorized traffic display statute. Ultimately, in the fourth confrontation, Faustin was cited by Sergeant W.P. Honer for violating the posting ordinance; however, this charge was dismissed in open court when the Denver city prosecutor realized that, because Faustin was merely holding the banner and not affixing it to anything, the posting ordinance was inapplicable to Faustin’s conduct. Thus, Faustin was never criminally tried.
In November 1998, Faustin filed this civil rights action in federal court pursuant to 42 U.S.C. § 1983. Both sides filed motions for summary judgment, and the district court granted summary judgment for Faustin. On the first appeal in Faustin I, we held that Faustin had no standing to seek prospective relief from the posting ordinance or to challenge the unauthorized traffic display statute on its face. 268 F.3d at 948-49. We also determined that the district court had not addressed Faus-tin’s separate challenge to Denver’s unwritten policy relating to expression on overpasses. Id. at 950. After determining that Faustin has standing to challenge this policy, we remanded for further proceedings to determine whether the policy violates the First Amendment. Id.
Upon remand, the district court considered supplemental briefing from the parties and concluded that Denver’s unwritten policy “to prohibit all speech or expressive activities on all highway overpasses located in the City and County of Denver” is unconstitutionally overbroad and vague. The court entered a judgment granting Faus-tin’s original motion for summary judgment and denying Denver’s original cross-motion. Denver appeals, urging us to reverse the district court’s judgment and order entry of summary judgment in Denver’s favor.
II. Standard of Review
We review the grant of summary judgment de novo. Axson-Flynn v. Johnson, 356 F.3d 1277, 1283 (10th Cir.2004). We also review the district court’s findings [1196]*1196of constitutional fact in a First Amendment claim and conclusions of law de novo. Hawkins v. City and County of Denver, 170 F.3d 1281, 1285 (10th Cir.1999). Because this decision implicates First Amendment freedoms, we perform an independent examination of the whole record in order to ensure that the judgment protects the rights of free expression. Id.
III. Discussion
A. Faustin’s claims
There are two types of First Amendment challenges that can be brought against a city policy, facial and as applied.1 A facial challenge considers the restriction as a whole, while an as-applied challenge tests the application of that restriction to the facts of a plaintiffs concrete case. See Hawkins, 170 F.3d at 1286, 1290. Facial challenges seek to vindicate not only individual plaintiffs’ rights but also those of all others who wish to engage in the speech being prohibited. See id. at 1286.
The district court, on remand, said that “plaintiff challenges this policy on its face” and considered only facial over-breadth and vagueness claims. Although it does not appear that Faustin asserted a facial challenge to the policy in her original complaint, Faustin did develop these facial challenges over the course of the litigation in the district court, both before and after our remand in Faustin I. Defendants never challenged the characterization of Faus-tiris developing claims as a facial challenge to Denver’s policy. Therefore, we hold that Faustin impliedly amended her complaint in the district court, and we will address the facial vagueness and over-breadth claims before us in this appeal. See Fed.R.Civ.P. 15(b) (“When issues not raised by the pleadings are tried by express or implied consent of the parties, they shall be treated in all respects as if they had been raised in the pleadings”); accord Green Country Food Mkt., Inc. v. Bottling Group, Inc., 371 F.3d 1275, 1280 (10th Cir.2004).2
B. Denver’s policy
The parties dispute the precise parameters of Denver’s unwritten policy and refused to stipulate to any single version of the policy at oral argument. The evidence before us on summary judgment presents a somewhat contradictory picture. In some instances, Defendants describe Denver’s policy broadly as prohibiting all expression on all overpasses.3 We also used regrettably imprecise language in Faustin I when we said that “closer analysis of the complaint, the motions for summary judg[1197]*1197ment, and the briefing before us indicates that Faustin also challenges defendants’ policy of prohibiting all expression on overpasses. Defendants admit this policy exists.” 268 F.3d at 950. Faustin emphasizes these “all expression” references and argues the policy therefore prohibits not just her protest activities but every kind of expression on overpasses, including bumper stickers on cars and conversations among pedestrians.4
However, Faustin has offered no evidence to prove Denver’s policy has ever reached beyond expression visible to the traffic below. To the contrary, the evidence that Faustin has submitted suggests that Denver has only restricted signage on overpasses — evidence that an animal rights protestor was previously arrested after displaying a large protest sign on a highway overpass in Denver.5 Indeed, except for intermittent references to “all expression,” Defendants have consistently articulated a more narrow version of the policy that prohibits only the display of signs and banners on overpasses.6 Indeed, Lt. Donald Fink himself demonstrated in an affidavit before the district court that “all expression on overpasses” is not nearly as inclusive as Faustin hypothesizes, where he stated:
It has been the uniform and consistent policy and practice of the Denver Police Department to prohibit all speech or expressive activities on all highway overpasses located in the City and County of Denver. Put another way, it is the Denver Police Department’s policy and practice to advise all persons who wish to engage (or are engaging) in speech activities at these sites that they cannot display banners or signs while at these locations, whether or not such signs or banners are attached to protective fencing.
(Emphasis added.)
The district court analyzed this case based only on this broadest “all expression” language. The court acknowledged that “the parties dispute the exact nature of defendants’ policy.” Nonetheless, relying on our discussion in Faustin I, the district court concluded that the “law of the case mandates that defendants” “uniform and consistent policy and practice ... is to prohibit all speech or expressive activities on all highway overpasses located in the City and County of Denver.”
However, although our language in Faustin I was unfortunately ill-defined, nothing in our prior opinion nor in the [1198]*1198factual record would support a finding of any policy that broad.7 Given the tortuous history of this litigation, caused in no small measure by the imprecision of the parties’ positions at various stages of this litigation, it would be a waste of judicial resources and an injustice to the parties to keep this controversy alive longer to obtain a further refinement of precisely what Denver’s policy is. We will simply conduct our review based on the broadest policy reasonably inferable from the record. Upon our de novo review of this evidence, we determine that the broadest possible policy that Denver could have had was a policy prohibiting all expressive conduct on overpasses that was visible to traffic below and potentially disruptive to that traffic on the underpass.8
C. Summary judgment
Before Faustin I, the district court considered cross-motions for summary judgment. Denver now appeals from the district court’s second grant of summary judgment in Faustin’s favor and its denial of Defendants’ cross-motion for summary judgment. Both sides urge us that summary judgment is the proper way to resolve this case.
“Summary judgment is proper if the evidence submitted by the parties, viewed in the light most favorable to the non-movant, indicates that there is no genuine issue as to any material fact and the moving party is entitled to judgment as a matter of law.” Smith v. City of Enid, 149 F.3d 1151, 1154 (10th Cir.1998) citing Fed.R.Civ.P. 56(c). “A fact is material if under the relevant substantive law it is essential to proper disposition of the claim.” Wright v. Abbott Labs., Inc., 259 F.3d 1226, 1231-32 (10th Cir.2001). Only material factual disputes can create a genuine issue for trial and preclude summary judgment. Hall v. Bellmon, 935 F.2d 1106, 1111 (10th Cir.1991). Therefore, although there is some dispute as to the scope of Denver’s policy, because we have assumed the broadest policy supportable by the record and still conclude that Faustin fails to establish that such a policy is either unconstitutionally broad or vague, this dispute becomes immaterial.
Where appropriate, we may grant summary judgment “in favor of any moving party we conclude is entitled to summary judgment on the record before us.” First Unitarian Church of Salt Lake City v. Salt Lake City Corp., 308 F.3d 1114, 1120 (10th Cir.2002). We may even grant summary judgment to a non-moving party if the “facts were fully developed at the summary judgment hearing so that the court of appeals can determine that the nonmoving party clearly was entitled to a [1199]*1199judgment as a matter of law ... [and] there is no procedural prejudice to the moving party.” Dickeson v. Quarberg, 844 F.2d 1435, 1444 n. 8 (10th Cir.1988) (quoting 10A Wright, Miller & Kane, Federal Practice & Procedure § 2720). Here, as our following discussion makes clear, we conclude that Faustin is not entitled to summary judgment but rather that on the facts most favorably interpreted for Faus-tin, as the opposing party in this regard, summary judgment should be granted to Defendants. See First Unitarian, 308 F.3d at 1120.
D. First Amendment analysis
We now turn to the substantive issues before us. Specifically, we must decide whether Denver has implemented a policy that is so overbroad or vague that its very existence violates the First Amendment. “Because facial challenges push the judiciary toward the edge of its traditional purview and expertise, courts must be vigilant in applying a most exacting analysis to such claims.” Ward v. Utah, 398 F.3d 1239, 1247 (10th Cir.2005).
1. Overbreadth
Under First Amendment over-breadth analysis, “[t]he showing that a law punishes a substantial amount of protected speech, judged in relation to the statute’s plainly legitimate sweep, suffices to invalidate all enforcement of that law, until and unless a limiting construction or partial invalidation so narrows it as to remove the seeming threat or deterrence to constitutionally protected expression.” Virginia v. Hicks, 539 U.S. 113, 118-19, 123 S.Ct. 2191, 156 L.Ed.2d 148 (2004) (emphasis in original) (quotation omitted). This “expansive remedy” is granted “out of concern that the threat of enforcement of an over-broad law may deter or ‘chill’ constitutionally protected speech” because “[m]any persons, rather than undertake the considerable burden (and sometimes risk) of vindicating their rights through case-by-case litigation, will choose simply to abstain from protected speech — harming not only themselves but society as a whole....” Id. at 119, 123 S.Ct. 2191 (citations omitted).
Overbreadth is “strong medicine,” and courts “employ[] it with hesitation, and then, only as a last resort.” West, 206 F.3d at 1367 (quotation omitted); see also Broadrick v. Oklahoma, 413 U.S. 601, 613, 93 S.Ct. 2908, 37 L.Ed.2d 830 (1973). The Supreme Court has noted that “there comes a point at which the chilling effect of an overbroad law, significant though it may be, cannot justify prohibiting all enforcement of that law — particularly a law that reflects legitimate state interests in maintaining comprehensive controls over harmful, constitutionally unprotected conduct.” Hicks, 539 U.S. at 119, 123 S.Ct. 2191 (quotation omitted). At that point, there are “substantial social costs created by the overbreadth doctrine....” Id. (emphasis added).
Accordingly, to prevail on an over-breadth challenge, a plaintiff must establish substantial overbreadth. Broadrick, 413 U.S. at 615, 93 S.Ct. 2908. “The over-breadth claimant bears the burden of demonstrating, from the text of the law and from actual fact, that substantial over-breadth exists.” Hicks, 539 U.S. at 122, 123 S.Ct. 2191 (internal quotations, alternations omitted); accord Ward, 398 F.3d at 1248.
In addition to being substantial, the overbreadth must also be real. Broadrick, 413 U.S. at 615, 93 S.Ct. 2908; see also West, 206 F.3d at 1367 (10th Cir.2000) (requiring plaintiff to establish a “realistic” chilling effect on the free speech rights of third parties not before the court who have been prevented from engaging in protected speech). The Supreme Court explained:
the mere fact that one can conceive of some impermissible applications of a [1200]*1200statute is not sufficient to render it susceptible to an overbreadth challenge. On the contrary, the requirement of substantial overbreadth stems from the underlying justification for the overbreadth exception itself — the interest in preventing an invalid statute from inhibiting the speech of third parties who are not before the Court.... In short, there must be a realistic danger that the statute itself will significantly compromise recognized First Amendment protections of parties not before the Court for it to be facially challenged on overbreadth grounds.
Members of the City Council of Los Angeles v. Taxpayers for Vincent, 466 U.S. 789, 800-01, 104 S.Ct. 2118, 80 L.Ed.2d 772 (1984) (footnote omitted).
Thus, in this ease, Faustin must establish that Denver’s policy actually sweeps within its prohibitions such a substantial amount of protected free speech in relation to its “plainly legitimate sweep” that the policy itself must be invalid on its face. Our analysis, thus, starts with assessing the policy’s “plainly legitimate sweep.” See Hicks, 639 U.S. at 120, 122 n. 3; Ward, 398 F.3d at 1247.
a. valid time, place, and manner restrictions
We apply the time, place, and manner doctrine to evaluate whether restrictions placed on expression in public fora violate the First Amendment.9 Hill v. Colorado, 530 U.S. 703, 731, 120 S.Ct. 2480, 147 L.Ed.2d 597 (2000); see also Wells v. City and County of Denver, 257 F.3d 1132, 1147 (10th Cir.2001). “This doctrine allows the government to place restrictions on expression so long as it does not regulate what is said, but merely regulates such matters as when, where, and how loud.” Ward, 398 F.3d at 1254 (alterations omitted) (quoting Rodney A. Smolla, 1 Smolla & Nimmer on Freedom of Speech § 8.36 (2004)).
There is no dispute in this case that Denver’s policy is content-neutral. Thus, the test is whether Denver’s policy is a reasonable restriction on the time, place, or manner of protected speech that is “narrowly tailored to serve a significant government interest, and leave open ample alternative channels of communication.”10 Perry Educ. Ass’n. v. Perry Local Educators’ Ass’n., 460 U.S. 37, 45, 103 S.Ct. 948, 74 L.Ed.2d 794 (1983).
Applying this analysis, we conclude that Denver may constitutionally restrict the displaying of signs or banners on highway overpasses. The state has a significant interest in traffic safety and in the avoidance of interference with official traffic control devices on highway overpasses. Schenck v. Pro-Choice Network, 519 U.S. 357, 376, 117 S.Ct. 855, 137 L.Ed.2d 1 (1997); Metromedia, Inc. v. City of San Diego, 453 U.S. 490, 507-508, 101 S.Ct. 2882, 69 L.Ed.2d 800 (1981); cf. Wells v. City and County of Denver, 257 F.3d 1132, 1148 (10th Cir.2001).11 Restricting signs [1201]*1201or banners on highway overpasses directed at the motorists below is narrowly tailored to achieve this interest. See Rock Against Racism, 491 U.S. at 798-800, 109 S.Ct. 2746. And as for ample alternative avenues for communication, the policy affects only highway overpasses; Denver rightly notes that there are “hundreds of miles of sidewalks and thousands of acres of parks and other public fora to present ... views and ideas to the public.”
Therefore, Denver may constitutionally restrict signs or banners on highway overpasses, and such speech is legitimately subject to regulation. Thus, this restriction on expression falls within the plainly legitimate sweep of the policy. As such, if the policy, as Denver argues, only ever restricts signs and banners, it is not unconstitutional. Even if one were to hypothesize that the Denver policy might reach some expression on the overpass that could not impede traffic flow on the highway, there is nothing in this record to suggest that any such speech would be substantial in comparison to the legitimately prohibited expressions. See Hicks, 539 U.S. at 122, 123 S.Ct. 2191; Ward, 398 F.3d at 1248.
b. real, substantial overbreadth
Both Faustin and the district court rely on speculation that a policy prohibiting “all expression on all overpasses” has an unconstitutional chilling effect because it could extend to prohibit any conversation on the overpass walkway, the distribution of literature on the walkway, signs facing away from the traffic below, and even vehicles traveling over an overpass adorned with letters, symbols, or bumper stickers.
Although we recognize that to some degree the conclusion that a speech restriction has a “chilling effect” on parties not before the court requires a bit of judicial extrapolation, in this case Faustin has presented no evidence and made no showing that Denver’s policy has ever been applied to prohibit any expression on overpasses other than signs or banners facing motorists below or that Denver’s policy has ever been so broadly interpreted by the public in a way that it has chilled any such speech. To the contrary, the only instances in the record of the policy’s actual application relate to signs and banners — as applied to Faustin, as applied to an animal rights protestor, and in the only specific law enforcement testimony regarding specific past enforcement of the policy.
We are simply unsatisfied that a policy that has never been applied or threatened against anything other than signs or banners facing traffic below will have any realistic danger of chilling the expression of parties not before the court — at least not as to any expression that could not be legitimately restricted under the “time, place, and manner” analysis. Thus, Faus-tin has failed to show any realistic chilling effect. Absent any showing in this regard, Plaintiff cannot prevail at summary judgment and Defendants are entitled to judgment as a matter of law. Therefore, we reverse the district court and grant summary judgment on this point in Denver’s favor.
2. Vagueness
“It is a basic principle of due process that an enactment is void for vagueness if its prohibitions are not clearly defined.” Grayned v. City of Rockford, 408 U.S. 104, 108, 92 S.Ct. 2294, 33 L.Ed.2d 222 (1972). There are two possible, and independent, reasons a statute may be impermissibly vague: “First, if it fails to provide people of ordinary intelligence a reasonable opportunity to understand what conduct it prohibits. Second, if it authorizes or even encourages arbitrary and discriminatory enforcement.” Hill v. Colora[1202]*1202do, 530 U.S. 703, 732, 120 S.Ct. 2480, 147 L.Ed.2d 597 (2002).
A plaintiff may challenge a statute as overly vague where the statute’s deterrent effect on legitimate expression is “both real and substantial” and the statute is not “readily subject to a narrowing construction by the state courts.... ” Young v. Am. Mini Theatres, Inc., 427 U.S. 50, 60-61, 96 S.Ct. 2440, 49 L.Ed.2d 310 (1976). Speculation and “hypertechnical theories as to what the statute covers” cannot create vagueness, especially when the statute is “surely valid in the vast majority of its intended applications.” Hill, 530 U.S. at 733, 120 S.Ct. 2480 (quotation omitted). “And while there is little doubt that imagination can conjure up hypothetical cases in which the meaning of terms will be in nice question, because we are condemned to the use of words, we can never expect mathematical certainty from our language.” Id. at 733, 120 S.Ct. 2480 (internal citations, alteration, quotation omitted). Finally, where a plaintiff knows the action in question violated the restriction, we have said that this “state of mind is inconsistent with any claim that the policy did not give ... fair warning....” West, 206 F.3d at 1368. In such a case, we will not conclude that the policy is unconstitutionally vague. Id.
In this case, Denver’s policy is not vague, either facially or as applied to Faustin. Faustin knew and had no doubt that the restriction applied to her, and such application is not unconstitutional. Again, Faustin’s claim fails because she has made no showing of “real and substantial” chilling of protected speech. Essentially, Faustin’s vagueness argument goes to the fact that the policy is unwritten. However, this fact is not fatal. See Lebron v. National Railroad Passenger Corp., 69 F.3d 650, 658 (2d Cir.1995) (determining unwritten policy prohibiting all political advertising was not unconstitutional). The policy clearly prohibits communication directed toward the traffic on the underpass, and thus it is not unconstitutionally vague.12
IY. Conclusion
For the foregoing reasons, we REVERSE the district court and REMAND with instructions to grant summary judgment in Defendants’ favor. No. 04-1025, Faustin v. City and County of Denver