As we know, the uniformly convex function is a generalization of the strongly convex function. However, is there any example that belongs to the former, not the latter? Many thanks in advance.
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- $f(x)=x^2$ is strongly convex.
- $f(x)=x^3$ on $[0,1]$ is uniformly convex but not strongly.
- $f(x)=e^x$ strictly convex but not uniformly convex.
- $f(x)=|x|$ is convex but not strictly convex.
Paresseux Nguyen
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Thanks a lot for these nice examples. BTW: Is there any example defined on R instead of on an interval as in 2? – kaienfr Jul 10 '22 at 19:00
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@kaienfr: Yes, there is. You just have to extend $f(x)=x^3$ with $x^2$ – Paresseux Nguyen Jul 10 '22 at 20:21