## New answers tagged convexity

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 ...

- 14.2k

9
votes

Accepted

### 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 ...

- 6,115

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,...

- 40.5k

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