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Results tagged with convex-geometry
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user 32507
A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...
6
votes
Accepted
Are the polyhedral cones the only examples of cones that remains closed when they are added ...
The radial cone of $C$ is defined via
$$
\mathcal R_C(x) := \bigcup_{\lambda > 0} \lambda ( C - x)$$
for all $x \in C$
and we can show
$$
\mathcal R_C(x) = C + \operatorname{span}(x),
$$
since $C$ is …
- 1,724
2
votes
Accepted
A converse question about the polyhedrality under linear mapping
I think we can argue as in https://mathoverflow.net/a/423284/32507 to answer the question in the affirmative.
Let $\mathcal R_K(x)$ be the radial cone of $K$ at $x$ (as defined in the other answer). F …
- 1,724
2
votes
Accepted
When are infimal convolutions contractions?
Here are just some thoughts. I think it is a matter of curvature, so let us assume that $\varphi$ and $\psi$ are smooth. Then, $y(x)$ solves the optimality condition
$$
\psi'(y(x)) = \varphi'(x - y(x) …
- 1,724
0
votes
On faces of polytopes
The set $K_A$ is essentially a polar of $A$. Indeed, we have
$$
A = \{ x \in \mathbb R^n \mid l(x) \ge t \; \forall (l,t) \in K_A\} =: B.$$
The inclusion "$\subset$" is clear and in order to check "$\ …
- 1,724