Inscribing a regular octahedron into polytopes

Inscribing a regular octahedron into polytopes

We prove that any simple polytope (and some non-simple polytopes) in R admits an inscribed regular octahedron.

Random inscribing polytopes

For convex bodies K with C2 boundary in Rd, we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

An Inscribing Model for Random Polytopes

For convex bodies K with C2 boundary in Rd, we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

Inscribing Smooth Knots with Regular Polygons

A knot K : S → R is said to be inscribed by a regular n-gon if there is a set of points x0, ..., xn−1 lying on K in a cyclic order, such that the distances ‖xi−1 − xi‖ between xi−1 and xi are the same for i = 1, ..., n, where xn = x0. Jon Simon asked the question of whether every smooth knot K is inscribed by a regular n-gon for all n. Up to rescaling we may assume that the length of K is 1. It...

Regular polytopes