All Questions
71 questions
4
votes
4
answers
536
views
Vertex-transitive polytopes in any dimension with any number of vertices?
Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I ...
- 51
10
votes
3
answers
537
views
Perimeter-halving center of a convex shape
Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ perimeter-halving
if half the perimeter lies to one side of $c$
(and so half to ...
- 151k
5
votes
0
answers
93
views
Which polytopes can be deformed while keeping their edge-lengths?
Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while
keeping its combinatorial type, and
keeping its ...
- 13.6k
21
votes
2
answers
1k
views
On convergence of convex bodies
Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...
- 21.8k
1
vote
2
answers
153
views
Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
- 5,979
6
votes
4
answers
2k
views
Delaunay triangulations and convex hulls
This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
2
votes
0
answers
103
views
Polytopes with large dihedral angles
The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$.
The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
- 13.6k
5
votes
0
answers
139
views
On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
- 5,979
3
votes
0
answers
52
views
Deformations that flatten small curvature
I'm trying to show that any 3-dimensional polyhedron with many vertices can be mildly deformed so that its vertices are no longer convexly independent. I suspect it suffices to look at a vertex with ...
- 31
9
votes
1
answer
370
views
Largest convex hull of a unit length path
What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
- 851
13
votes
1
answer
430
views
Detecting a hidden convex body with line probes
Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...
- 151k
4
votes
1
answer
124
views
Convex caps with prescribed edges and curvature
Let $G$ be the edge graph of a convex subdivision of a convex polygon $P$ in the plane. I would like to construct a convex polyhedral cap $C$ (with zero boundary values) over $P$ whose edges project ...
- 7,194
3
votes
1
answer
205
views
How to show it is contained in a convex hull?
There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have
$$
\mathcal{H}(F_i)\...
- 43
1
vote
2
answers
1k
views
Is there always a parallelogram cross-section of parallelepiped contained in the smallest box
Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...
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
0
answers
415
views
Find the intersection between two convex hulls, in this specific case
We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$.
Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
- 233
7
votes
3
answers
377
views
Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
- 151k
5
votes
0
answers
135
views
What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?
Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...
- 13.4k
4
votes
1
answer
333
views
n-simplex in an intersection of n balls
Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
- 195
4
votes
1
answer
203
views
Covering a convex body with its smaller homothetic copies
Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...
- 285
1
vote
1
answer
176
views
Helly's number from biconvex functions
Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \...
- 135