Volumes in High Dimension

Abstract

We begin with comparing the volume of the n-dimensional cube with the volume of the unit ball inscribed in it, in order to realize that volumes of “familiar” bodies behave quite differently in high dimensions from what the 3-dimensional intuition suggests. Then we calculate that any convex polytope in the unit ball B ⁿ whose number of vertices is at most polynomial in n occupies only a tiny fraction of B ⁿ in terms of volume. This has interesting consequences for deterministic algorithms for approximating the volume of a given convex body: If they look only at polynomially many points of the considered body, then they are unable to distinguish a gigantic ball from a tiny polytope. Finally, we prove a classical result, John’s lemma, which states that for every n-dimensional symmetric convex body K there are two similar ellipsoids with ratio $\sqrt {n}$ such that the smaller ellipsoid lies inside K and the larger one contains K. So, in a very crude scale where the ratio $\sqrt {n}$ can be ignored, each symmetric convex body looks like an ellipsoid.