Articles of polytopes

Edges of a permutohedron

Consider a permutohedron $P_n$ (this is a polytope which is a convex hull of $n!$ points, which are obtained from $(1,2,…,n)$ by all possible permutations of coordinates). I have to prove the following: 2 vertices are connected with an edge iff their coordinates differ by transposition of 2 numbers which differ by 1. I have […]

Prove that an edge of a polyhedron is a line segment

Preliminaries Let us define $\mathbb{N}_n=[1,n]\cap\mathbb{N}^+$ for all $n\in\mathbb{N}^+$. $\ $ A polyhedron is an intersection of halfspaces, i.e., a set of the form $\{x\in\mathbb{Q}^n\mid Ax\le b\}$ for some matrix $A\in\mathbb{Q}^{m\times n}$ and column vector $b\in\mathbb{Q}^m$. $\ $ Let us say that a face of a polyhedron $P$ is a set of points of the form […]

Exposed point of a compact convex set

I’m trying to show that given a compact convex set $K$ in $R^d$, there must be at least one exposed point (where $v$ is exposed if there exists a hyperplane H such that $H \cap K = \{v\}$ . This is a homework problem, but I’m totally stuck and looking for a hint. I know […]

Explain `All polyhedrons are convex sets´

My teacher in course in Mat-2.3140 of Aalto University claims that ‘All polyhedrons are convex sets’ here. This premise was in a false-or-not-problem ‘The feasible set of linear integer problem is polyhedron’. You can see below the screenshot of the solution. Wikipedia shows nonconvex polyhedrons such as orthogonal polyhedron here. What should I now believe? […]

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 simplex $P$ with vertices $V_{0..N+1}$ which would contain $S$? A test of $S_j$ being interior to the simplex and not on or outside of […]

Counts of certain types of faces of the $CQHRL_d$ polytope family

This is essentially a counting question, with motivation being supplied by a particular polytope family called crenellated quad hyper-rope loops of dimension $d$ ($CQHRL_d$) ($d$ $\ge$ $5$) and certain faces of them. In particular, the question can be answered just by reference to the progression of diagrams (the first few shown below) which suffice to […]

Strong equidistribution of points on the n-sphere

The vertices of a Platonic solid are equally distributed on its circumscribing sphere in a very strong sense: each of them has the same number of nearest neighbours and all distances between nearest neighbours are the same. It seems clear to me that the Platonic solids also provide the only examples for such equidistributed arrangements […]

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite Element theory concerning triangles on MSE. The concept of isoparametrics is introduced herein. A reference to the same theory is found in: […]

How can I visualize a four-dimensional point inside a Schlegel diagram of a tesseract?

I would like to draw a Schlegel diagram of a tesseract to visualize via a Cartesian coordinate system inside the tesseract the symmetry of some four-dimensional points located in a range of integer values no longer than the half of the length of the side of the tesseract. (The questions are at the end of […]