13
votes
0answers
339 views

Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?
7
votes
0answers
91 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
0
votes
1answer
87 views

A question about rational convex cone

Edit: Fix a lattice $N = \mathbb{Z}^n$ and let $N_{\mathbb{R}} = N \otimes \mathbb{R}$. Let $C(S)$ be a strongly convex rational cone in $N_{\mathbb{R}}$ generated by a finite set $S \subset N$, with ...
1
vote
1answer
200 views

non-convex Polytope definition

I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$, we can define a convex polytope in the following way: $$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, ...
2
votes
1answer
131 views

coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...
17
votes
2answers
779 views

Placing points on a sphere so that no 3 lie close to the same plane

Motivation I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...