The given convex body can be approximated by a sequence of nested bodies, eventually reaching one of known volume (a hypersphere), with this approach used to estimate the factor by which the volume changes at each step of this sequence.

We concentrate on two main topics: Geometric properties: volume and surface area, mixed volumes, and quer-massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes.

A convex body in Rn is a compact convex set with non-empty interior. A body K is called symmetric if x ∈ K =⇒ −x ∈ K. Convex hull of N points x1,...,xN. n conv(x1,...,xN) = {y ∈ R : y = λ1x1 +...+λNxN, λi ≥ 0, Xλi = 1}. conv(x1,...,xN) = n {y ∈ R : y = λ1x1 +...+λNxN, λi ≥ 0, Xλi = 1}.

We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape. Calculating the volume of geometric objects is a major topic in analytic geometry. In high school, we humbly touch it when we study two- and three-dimensional shapes.

We will give a randomized, polynomial time algorithm for approximating the volume of a convex body, given a separation oracle. The presentation roughly follows the original Dyer, Frieze, Kannan paper, and gives a

use vol(K) to represent the volume of a convex body K, and B(x,R) to represent the ball with radius R and center x. Like the original multiphase Monte-Carlo algorithm, our algorithm consists of three

(d) vK is convex. In fact, vK is strictly log-convex on int(K∗). (e) vK is infinitely often differentiable on int(K∗). Proof (a) For all x =0, one has Hx ⊇ x −1 Bn. Hence, vK (x) ≥ voln K ∩( x −1Bn) = voln x −1 (K ∩Bn) = x −n btv(K). (b) Let x ∈ Rn.ThesetK ∩ …

In particular, the notions of mixed volume and mixed area measure arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered here in detail. The author presents a comprehensive introduction to convex bodies, including full proofs for some deeper theorems.

5 Convex Bodies: Mixed Volumes and Inequalities 173 A general convex body is neither polyhedral nor smooth. Take for example a Cantor set in the circle and form its convex hull. Bounded convex sets in Rd are measurable, even in the Jordan sense. We denote

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coe cients of this polynomial, called mixed volumes.