Or as I call it, “Ancient Greek Victory Headband”
I believe the correct word is tainia (ταινία), as opposed to diadema (διάδημα); diadema seems to be used frequently with the more ornate headwear worn by women (similar to a tiara). A tainia of red wool was often placed around the head of the victor of an event at an Ancient Greek game.
The “Primary Attributes of Victor Statues” chapter of  has a passage that supports this interpretation of a tainia: “Pausanias says that the victor Lichas of Sparta was scourged by order of the umpires at Olympia for having set the tainia on the head of his victorious charioteer. This is sufficient evidence that it was not a mere toilet article, but rather a part of the official prize of victory.” Other examples abound. In Pausanias’ Description of Greece 6.20.19  (2nd century AD), we find more evidence for the tainia as a ribbon that is used to “crown” a victor: “ἐπὶ δὲ νύσσης μιᾶς Ἱπποδαμείας ἐστὶν εἰκὼν χαλκῆ, ταινίαν τε ἔχουσα καὶ ἀναδεῖν τὸν Πέλοπα μέλλουσα ἐπὶ τῇ νίκῃ.” (“On one turning-post is a bronze statue of Hippodameia carrying a ribbon, and about to crown Pelops with it for his victory.”) In his Republic  (360 BC), Plato writes”For the prize of victory that they win is the salvation of the entire state, the fillet that binds their brows is the public support of themselves and their children — they receive honor from the city while they live and when they die a worthy burial.” In Xenophon’s Symposium  (360 BC), it’s clear that tainiai are synonymous with victory: “ἡ μὲν δὴ παῖς καὶ ὁ παῖς κρύφα ἀνέφερον. ὁ δὲ Σωκράτης ἐν τούτῳδιέπραττε τόν τε λύχνον ἀντιπροσενεγκεῖν τῷ Κριτοβούλῳ, ὡς μὴἐξαπατηθείησαν οἱ κριταί, καὶ τῷ νικήσαντι μὴ ταινίας ἀλλὰ φιλήματαἀναδήματα παρὰ τῶν κριτῶν γενέσθαι.” (“So the maiden and the lad turned in the ballots secretly. While this was going on,Socrates saw to it that the light should be brought in front of Critobulus, so that the judges might not be misled, and stipulated that the prize given by the judges to crown the victor should be kisses and not ribbons.”)
Virgil’s Aeneid  gives insight to the color of the tainia: “Bearing such gifts, th’ exultant victors onward moved, each brow bound with a purple fillet.” But the word translated as purple, puniceis, is a form of puniceus, which is defined as “reddish, red, purple-colored” by An Elementary Latin Dictionary  and has links to the Greek word for Tyrian purple, phoînix. We can then presume that the tainia here described are similarly colored.
Presumaby, a white tainia was used as symbols of kingship, but this seems to be frequently stated without support. Many kings are depicted with a fillet around the head, such as the depiction of Perseus of Macedon in  and Areus I of Sparta , but as yet, I have not found evidence for the color. The Charioteer of Delphi (478 or 474 BC) statue shows a tainia with an embattled pattern .
It is very interesting that a victor should receive a red tainia, and a king would presumably wear a white one. The embattled decoration on the charioteer’s tainia is also interesting. Put into an SCA context, a white tainia with an embattled pattern seems like an appropriate, and historically inspired, alternate coronet-style within the SCA for an Ancient Greek persona having at least a County.
 W. W. Hyde. Olympic victor monuments and Greek athletic art. Carnegie Institution of Washington, 1921.
 Pausanias Description of Greece. Internet: http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:greekLit:tlg0525.tlg001.perseus-eng1:6.20.19 [March 6, 2017].
 Plato Republic. Internet: http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:greekLit:tlg0059.tlg030.perseus-eng1:5.465d [March 6, 2017].
 Xenophon Symposium. Internet: http://www.perseus.tufts.edu/hopper/text?doc=Perseus:abo:tlg,0032,004:5:9&lang=original [March 6, 2017].
 Virgil Aeneid. Internet: http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:latinLit:phi0690.phi003.perseus-eng2:5.244-5.285 [March 6, 2017].
 Charlton T. Lewis, An Elementary Latin Dictionary. http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0060%3Aentry%3DPuniceus [March 6, 2017].
 Coin of Perseus of Macedon. Internet: https://en.wikipedia.org/wiki/Perseus_of_Macedon#/media/File:Perseus_(1).jpg [March 6, 2017].
 Coin of Areus I. Internet: https://en.wikipedia.org/wiki/Areus_I#/media/File:Areus_I_King_of_Sparta.jpg [March 6, 2017].
 Charioteer of Delphi close up. Internet: https://en.wikipedia.org/wiki/Charioteer_of_Delphi#/media/File:Vognstyreren-fra_Delfi2.jpg [March 6, 2017].