Sub-linear functions over the real line.
Are these equivalent to linear functions, when continuous and F(0)=0?
Sublinear F often satisy sub-additivity (A) and a form of Homogeneity (B)
(A)-sub-additive forall (x,y)\in dom(F): F(x +y)
ie these will have more then one fixed pt with F(1)=1 and
if strictly increasing, continuous, F(0)=0,sub-linear models which midpoint convex, these will be convex and super-additive over the positive reals; but they are also sub-additive. Not to mention that they have many fixed pts
see https://math.stackexchange.com/questions/2247721/strictly-monotonic-continuous-sub-linear-functions-with-f1-1-any-real-difference
Using the definition of star convexity at 0, with F(0)=0, , and F(0.5)=.5)
0.5=F(0.5)=F(1/3*0+ 2/3*0.75)\leq (1/3F(0)+2/3F(0.75)=2/3F(0.75); ie 0.5\leq 2/3F(0.75)
[2/3*F(0.75)]\geq 0.5 implies F(0.75)\geq 3/2* 0.5=0.75, that is without using symmetry given star convexity for all t but only at 0, so that x cannot vary from 0, we can still choose y=1, and freely choose t, so that, using F(0)=0, F(1)=1, t=3/4
F(1/4\times 0 +3/4\times y), let y=1 , F(3/4)=F(1/4* 0 +3/4* 1)\leq1/4F(0)+3/4*F(1)=3/4
F(0.75)\geq 0.75 F(0.75)]\leq 0.75 F(0.75)=0.75) and F(x)=x for all x>0.5 \in [0,1](maybe all), The right half of the function will be linear. I presume it just will be linear. In fact I presume that even with star convexity at 0 with strict monotonicity and continuity and three fixed points F(1)=1 and F(0.5) and F(0)
see https://math.stackexchange.com/questions/2265396/can-a-strictly-increasing-convex-function-f-meet-a-line-segment-in-3-places-w
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