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7 votes

Existence of an open convex set

As shown by Saúl RM, the answer is negative. On the other hand, for absolutely convex sets (i.e., in addition to convexity we have $tx\in K$ for all $x\in K$ and $t\in [-1,1]$, if $0\in K$ this is the ...
9 votes

Existence of an open convex set

A convex $O'$ need not exist: a counterexample is given by setting $K=[-1,1]\times[0,2]\subseteq\mathbb{R}^2$ and $O=\{(x,y)\in K;y>x^3\}$. Indeed, any open $O'$ with $O'\cap K=O$ would contain ...
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2 votes

Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?

Note that the function $f\colon x\mapsto -c(x_0,x)$ is $c$-convex. Further for $c(x,y)=|x-y|^2_g/2$, the function $f$ is not convex in a neighborhood of $x_0$. So the answer is "no". However,...

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