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Are finite convex functions on $\mathbb R ^d$ continuous on $\mathbb R ^d$ when the underlying topology is different?

This question and your other recent question are both answered by the observation that one can easily define new metrics on $\mathbb{R}$ that are totally unconnected with the algebraic and order ...
Joel David Hamkins's user avatar
2 votes
Accepted

Do separable cubic constraint and separable quartic constraint SOCP presentable?

In general, for every rational number $r=p/q>1$ the inequality $t\geq x^r$ is convex on the set where $x\geq 0$ and it can be modeled using second-order cones. The model depends on $p,q$ and gets ...
Michal Adamaszek's user avatar
1 vote

Is every face exposed if all extreme points are exposed?

It is also possible to construct counterexamples from the idea that every extreme point of the convex hull of any subset of the unit sphere $S^2$ is an exposed point. Placing a convex set $C$ inside ...
Stephan's user avatar
  • 31
2 votes

Faces of the intersection of convex sets

I found a sufficient condition that guarantees that the answer to your question is yes: If the relative interior of $F$ is not empty, then $F$ is the intersection of a face of $K_1$ and a face of $K_2$...
Stephan's user avatar
  • 31
5 votes
Accepted

Integral means vs infinite convex combinations

No. Let $(X, \cal A, \mu)$ be $[0,1]$ with Lebesgue measure. Let $E = L^2[0,1]$ with inner product $\langle \alpha,\beta\rangle := \int \alpha(t)\overline{\beta(t)}\;dt$. Define $f : [0,1] \to L^2[0,1]...
Gerald Edgar's user avatar
  • 40.2k
4 votes

Integral means vs infinite convex combinations

I don't think so. Consider the functions $f(x,y)=e^{ixy}, -1<x<1, y\in \mathbb{R}$. Then, $$ \int_{-1}^1 f(x,y) \frac{dx}{2} = \frac{\sin(y)}{y}. $$ The question is if this is representable as $...
an_ordinary_mathematician's user avatar

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