INVESTIGATION IN THE CONCENTRATION OF VOLUME PHENOMENON

Abstract

Geometry in very high dimensions is full of surprises, many of the properties of high dimensional geometric objects could contradict with our intuition in 3-D Euclidian Space. The volume concentration phenomenon can easily be shown by the use of Lebesgue’s measure and probabilistic measure. Both measures can be useful in different scenarios. In this paper, we provide an introduction to Convex Geometry in high dimensions and discuss the idea of using probabilistic measure. We will then give example to visualize them. We will start with direct proof of the volume concentration of n-ball by using Gaussian integration. We will provide a proof for the Isoperimetric Inequality by proving the Brunn-Minkowski’s Inequality using Prekopa-Leindler’s Inequality. Then, we will investigate 1-Lipschitz function on n-Sphere, where we would find the volume concentration of a flat function on a sphere. Finally, we will show the volume concentration of a sphere on its equator by using a probabilistic arguments, where we can show that the mass of a n-Sphere is distributed normally along any axis.