# Convex hull of randomly chosen points from a polytope

• Rex Dwyer
• Ravi Kannan
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 269)

## Abstract

Suppose we pick independently n random points X1, X2,..., Xn from a d dimensional polytope P. (i.e., Xi 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 X1, X2,... X n . The following questions arise naturally:
1. 1)

What is the value of V n the expected ratio of the volume of P\E n to the volume of P?

2. 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.

## Keywords

Grid Point Convex Hull Extreme Point Half Space Adjacent Vertex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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