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Results tagged with convex-polytopes
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user 4354
Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
4
votes
Accepted
Zoll Flat Finsler tori and convex bodies on a starry night
No such a set is not always a polytope. Consider the convex hull of the set of points of the form $(1/n.1/n^2)$ and $(0,0)$ in $\mathbb R^2$. Its boundary is a union of infinitely many segments with r …
- 32.4k
25
votes
Accepted
Can all convex polytopes be realized with vertices on surface of convex body?
Yes, there is such a body. Actually there is one very close to the standard unit ball and containing disjoint representatives of each combinatorial type (but these representatives are very small).
In …
- 32.4k
17
votes
Accepted
Do random projections (approximately) preserve convexity?
Yes, if the convex body is "sufficiently round". If it is not, the resulting "closeness" to the boundary of a convex set is in absolute terms rather than relative. I don't know whether it can be impro …
- 32.4k
13
votes
Accepted
Isometric embedding a convex cap to render its boundary planar
Yes the polyhedral analog is true. Just consider the doubling of $C$, i.e., attach an isometric copy $C'$ of $C$ along the boundary, and apply Alexandrov's embedding theorem to the doubling. The commo …
- 32.4k