Random ball-polyhedra and inequalities for intrinsic volumes

Abstract

We prove a randomized version of the generalized Urysohn inequality relating mean width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.

Variance estimates for random disc-polygons in smooth convex discs

“…The notion of convexity arises naturally in many questions where a convex set can be represented as the intersection of equal radius closed balls. As recent examples of such problems, we mention the Kneser-Poulsen conjecture; see, for example, [7]- [9], and inequalities for intrinsic volumes in [22]. A more complete list can be found in [10], for short overviews, see also [15], [16], and [18].…”

On the intrinsic volumes of intersections of congruent balls

“…We note that r-ball bodies and r-ball polyhedra have been intensively studied (under various names) from the point of view of convex and discrete geometry in a number of publications (see the recent papers [2], [14], [16], [17], [19], and the references mentioned there). In particular, the following Blaschke-Santaló-type inequalities have been proved by Paouris and Pivovarov (Theorem 3.1 in [20]) as well as the author (Theorem 1 in [7]) for r-ball bodies in E d . Let A ⊂ E d , d > 1 be a compact set of volume V d (A) > 0 and r > 0.…”

“…On the one hand, the extended isoperimetric inequality (see for example, (1.1) in [20]) yields Next, notice that (47) for conv r P = S r,r0,2 and P r = L r,r−r0,2 yields…”

Randomized Isoperimetric Inequalities

“…Remark. The latter theorem is also true when V j is replaced by a function which is monotone with respect to inclusion, rotation-invariant and quasi-concave with respect to Minkowski addition; see [58]. The latter can be compared with the following result for the convex hull of unions of Euclidean balls.…”