A Polynomial Number of Random Points does not Determine the Volume of a Convex Body

TL;DR

This paper proves that even with a polynomial number of uniformly random points, it is impossible to reliably approximate the volume of a convex body in high-dimensional space.

Contribution

It establishes a fundamental limitation on volume estimation algorithms based on polynomially many random samples.

Findings

No polynomial-sample algorithm can approximate volume within a constant factor with high probability.

The result holds for convex bodies in high-dimensional spaces.

It highlights intrinsic computational hardness in volume estimation from random points.

Abstract

We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in R^n, can approximate the volume of the body up to a constant factor with high probability.