Convex hull of randomly chosen points from a polytope

Abstract

Suppose we pick independently n random points X ₁, X ₂,..., X _n from a d dimensional polytope P. (i.e., X _i are independent identically distributed random variables each with density = 1/volume(P) in P and 0 outside.) Let E _n be the convex hull of X ₁, X ₂,... X _n . The following questions arise naturally:

• 1) What is the value of V _n the expected ratio of the volume of P\E _n to the volume of P?

• 2) What is the expected number of extreme points of the polytope E _n ?

We show an upper bounds of C(P)/n(log n)^d+1 on V _n and C(P)(log n)^d+1 on M _n where C(P) is a constant that depends only on P (not on n). In both cases elementary arguments will only give a bound that replaces the power of log n by a power (less than one) of n. Previously, similar results were known only for the case of d=2. (Buchta (1984) and Rényi and Solanke (1963, 1964)). There has been substantial amount of work on the problem for spheres as well as for other quantities depending on E _n in two dimensions. (see for example W.M. Schmidt (1968), G.Buchta, J.Müller and R.F.Tichy (1985), P.M.Gruber (1983) and I.Bárány and Z.Füredi (1986)) In case the polytope P has at least one vertex with exactly d adjacent vertices, we prove lower bounds of d(P)(log n)^d−1 /n on V _n and d(P)log n)^d−1 on M _n .

Using the bounds, we are able to show that certain simple divide and conquer algorithms for finding the set of all extreme points have good sequential (linear time) and parallel (polylog time) complexitites in the expected case when the points are chosen at random independently from a polytope in a fixed number of dimensions.

The results are based on a natural notion of centrality which we introduce for convex sets.

Preliminary Version

Supported by NSF Grant ECS 8418392

Supported by NSF Grant ECS 8418392 and the Alexander von Humboldt Stiftung, Bonn