Iis there any discrete analogue m of star convexity at 0' star convexity at 0',?
That is, in the same way that midpoint convexity can be seen to be a discrete analogue of convexity (and entails it under certain regularity assumptions, continuity etc)?
For example, where F:[0,1] \to [0,1],
star convexity at 0 :
"\for all x \in dom (F)=[0,1],:(\forall t \in [0,1]) :"F(t * x_0 +(1-t) * x )
In the same way that midpoint convexity is the non continuous analogue that often renders a function continuous and convex? I presume that
mid- star convexity at 0 does not generalize to star convexity at o under continuity
(ie if midpoint convex function F, F:[0,1]\to [0,1]
F(0)=0 F(1)=1
is monotone increasing on its domain/lebesgue measurable function and continuous at 1; is generally convex and continous
such that if
if F is made continuous and satisifes the analogue (which is clearly not F(1/2x)
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