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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
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
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