All Questions
8 questions
13
votes
0
answers
573
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What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
- 3,068
7
votes
3
answers
412
views
Average caliper diameter (mean width) of a polyhedron
Define the caliper diameter of a polyhedron as follows:
Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...
- 101
20
votes
4
answers
950
views
The limit of edge-midpoint convex polyhedra
Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$,
replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$.
Continuing this process, we obtain a ...
- 151k
13
votes
0
answers
406
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?
- 131
14
votes
0
answers
479
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 ...
- 532
2
votes
1
answer
171
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 ...
- 14k
17
votes
2
answers
982
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 ...
- 1,314
6
votes
1
answer
544
views
Isometric embedding a convex cap to render its boundary planar
I would like to know if there is a polyhedral analog to this beautiful
theorem of Hong:
Theorem 11.0.1.
Any smooth positive disk $(\bar{D},g)$ with a positive geodesic
curvature along $\partial ...
- 151k