Volume, polar volume and Euler characteristic for convex functions

Expected volume and Euler characteristic of random submanifolds

In a closed manifold of positive dimension n, we estimate the expected volume and Euler characteristic for random submanifolds of codimension r ∈ {1, . . . , n} in two different settings. On one hand, we consider a closed Riemannian manifold and some positive λ. Then we take r independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenv...

A Volume Inequality for Polar Bodies

A sharp affine isoperimetric inequality is established which gives a sharp lower bound for the volume of the polar body. It is shown that equality occurs in the inequality if and only if the body is a simplex. Throughout this paper a convex body K in Euclidean n-space Rn is a compact convex set that contains the origin in its interior. Its polar body K∗ is defined by K∗ = {x ∈ R : x · y ≤ 1 for...

2 00 6 Average volume , curvatures , and Euler characteristic of random real algebraic varieties

We determine the expected curvature polynomial of random real projec-tive varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective varieties is found. This considerably extends previously known results on t...

On Volume Product Inequalities for Convex Sets

The volume of the polar body of a symmetric convex set K of Rd is investigated. It is shown that its reciprocal is a convex function of the time t along movements, in which every point of K moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the Lp-Blaschke-Santaló inequality for two-dimensional convex sets.

Cover for Volume.13, No.3