I have seen this condition being used for midpoint convexity:
Midpoint convexity. A set C is midpoint convex if whenever two points a,b are in C, the average or midpoint (a + b) is in c.; I have seen this in other documents, such as that if f(x) and f(y) are defined, then there exists some F(c), such that F(c)=F(x)/2+F(y)/2; such that perhaps x
Unless I am mis-using it; it seems implausibly strong a constraint; I can, it appear get all the way from jensens equality (if not something stronger, ie reading the numbers even factors of 3 off the table; and essentially something akin to linearity before i even use continuity). It appears that my symmetry condition (and only half of it, effectively entails outright jensens equality, on the whole structure given the first half- jensens equality, and even then, when only applied to half (less than half of the structure)
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