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The necessary condition for the polychoron to be regular (with Schläfli symbol {p,q,r}) and finite is cos(pi/q)<sin(pi/p)sin(pi/r). Sufficiency can be established by ...

Each point in the convex hull of a set S in R^n is in the convex combination of n+1 or fewer points of S.

There are a number of attractive polyhedron compounds consisting of ten octahedra. These compounds will be implemented in a future version of the Wolfram Language as ...

A real-valued function g defined on a convex subset C subset R^n is said to be quasi-concave if for all real alpha in R, the set {x in C:g(x)>=alpha} is convex. This is ...

Connectivity properties obey the following hierarchy: convex => star convex => pathwise-connected => connected.

Except for convex polygons, every simple polygon has at least one mouth.

A polynomial is called logarithmically concave (or log-concave) if the sequence of its coefficients is logarithmically concave. If P(x) is log-convex and Q(x) is unimodal, ...

Let a set of vertices A in a connected graph G be called convex if for every two vertices x,y in A, the vertex set of every (x,y) graph geodesic lies completely in A. Also ...

Given a simplex of unit content in Euclidean d-space, pick n>=d+1 points uniformly and independently at random, and denote the expected content of their convex hull by ...

A plane figure consisting of four points, each of which is joined to two other points by a line segment (where the line segments may intersect). A quadrangle may therefore be ...