Many explanations have been proposed for the function(s) of the POFs. In Equus at least, one possible cause (not "function") is the "caveing-in" of the malar region when it is no more "sustained" by cheek teeth because they are too worn.
As example, here is the skull of an E. caballus found (...) >suite
The scatter diagram SD-6 shows that (as far as I know) all Old World Hipparion skulls belong to Group B except H. prostylum (Lubéron LY 592) which has the same proportions as Group A-8 ( Ratio diagram A-8 and H. prostylum).
H. hendeyi (Langebaanweg E 22187), although larger, resembles (...) >continue
From the chronology of the sites (probably outdated table), it appears that the Group A is very old: Thomson, Echo Quarry, Trinity River. Even the brachyodont Mesohippus, Miohippus (Meso-Mio), Hypohippus, Megahippus and Parahippus (Hypo-Mega-Para) have the same shortness of the naso-incisival (...) >continue
The correlation between length of the cheek teeth P2-M3 and preorbital bar length (Scatter diagram 5) does not follow the same regression line for Cormohipparion (all morphs) and other archaic Northamerican equids. >continue
It seems that the main axial proportions of the skull are independent from the existence and/or type of POF.
For instance, on the ratio diagram of Cormohipparion A (A-1), the skull 42449 from Kepler looks like a typical Cormohipparion A although it has no POF.
The ratio diagram A-2 shows (...) >continue
Skull proportions and size of the preorbital bar (32)
1. Compared to many other ancient Northamerican equids (SD-2) Scatter diagram 2, all ‘typical’ Cormohipparion A and B (morphs 1) have a longer bar (32) relative to the length of the NIN (30).
2. The separations is, however, not so sharp (...) >suite
Unclassified.
Five skulls remain unclassified using this kind of ratio diagram (Unclassified 1).
Unclassified fragments. Fifteen other skulls were too fragmentary to be studied by this kind of ratio diagram (Unclassified 2). >suite
Dinohippus and Pliohippus
Dinohippus 11724, 87201, and Pliohippus 12998 have similar proportions (Dino-Plio 1) but only the last one has a well developped POF.
Five other skulls are all different (Dino-Plio 2). >suite