-1
votes
1answer
39 views

If we inscribed all the 6 regular convex four-dimensional polytopes in a sphere, which one would have the highest volume?

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). But what about for the 6 regular convex ...
0
votes
0answers
28 views

What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
0
votes
0answers
18 views

Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
1
vote
0answers
20 views

Is the volume of a convex polytope efficiently computable from the vertices

Is there an efficient method to compute the exact volume of a bounded full-dimensional convex polytope, given the coordinates of its vertices (V-representation)?
3
votes
1answer
31 views

Graph of polytope and hyperplane

Suppose that $P$ is a compact and convex polytope in $R^d$ and let $G$ be the graph of $P$ ($V(G)$ are the vertices of $P$ and $E(G)$ are the $1$-dimensional faces - for example polyedral graphs are ...
2
votes
0answers
113 views

Polytopes inside polytopes

I would like to share some facts about polytopes and how to check whether a polytope is a subset of some other polytope. I will provide some methods and I would like to know whether there exist other ...
3
votes
2answers
65 views

Need a way of computing the vertices of intersection of two simplices

I have two simplices $\Delta_1, \Delta_2$ defined as: The first simplex, $\Delta_1$, is the set of points defined as follows: $$\Delta_1 = \left\{\sum\theta_iu_i, \theta_i >= 0, ...
2
votes
1answer
110 views

Is the polar set of convex Polytope also Polytope

Let $P$ be a convex polytope. How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope? where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ . $Thanks$
2
votes
1answer
63 views

Are all polytopes also convex hulls?

It seems, at least in the 2-D case, that all polytopes are going to be convex. Does this hold if the dimensions are increased?
1
vote
1answer
103 views

Convex Hull in Hierarchy Structure

As a beggining to convex hull algorithms lecturer introduced the structure which it's called "Hierarchy Structure". Hierarchy Structure: in every given convex hull there is a maximum size convex hull ...
1
vote
1answer
72 views

Can i get every face of a polytope by taking a facet (of a facet (of a facet (…))) of the polytope?

Let $P$ be a polytope, i.e. a convex subset of a finite-dimensional real vector space with finitely many extreme points. Let $F$ be a proper face of the polytope, i.e. a subset $F \subset P$ such that ...
4
votes
1answer
68 views

A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
0
votes
0answers
144 views

Polytope as the intersection of closed half-spaces

I am stuck at a problem which looks very simple but which I still cannot prove. Let $P$ be a $d$-polytope. Say that $P$ can be represented as the intersection of a given finite set of closed ...
3
votes
1answer
153 views

Dual of a polytope — intersection of infinitely many halfspaces?

Let $S \subseteq \mathbb{R}^d$ be a $d$-dimensional convex set (i.e. $\exists d+1$ affinely independent points in $S$). Let the origin of the coordinate system lie in the interior of $S$ and let: ...
5
votes
0answers
199 views

How does one compute a bounding simplex for a set of points?

An algorithm I've implemented to tessellate an N-dimensional space requires starting with a bounding N-simplex. Given a set of $k$ points $S_{0..k-1} \subset R^N$ is there a procedure to find a ...