# Search for:All records

Creators/Authors contains: "Werner, Elisabeth_M"

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

1. Abstract

Using a natural representation of a 1/s-concave function on$${\mathbb {R}}^d$$${R}^{d}$as a convex set in$${\mathbb {R}}^{d+1},$$${R}^{d+1},$we derive a simple formula for the integral of itss-polar. This leads to convexity properties of the integral of thes-polar function with respect to the center of polarity. In particular, we prove that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions fors-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.

more » « less
2. Abstract

Consider two half-spaces$$H_1^+$$${H}_{1}^{+}$and$$H_2^+$$${H}_{2}^{+}$in$${\mathbb {R}}^{d+1}$$${R}^{d+1}$whose bounding hyperplanes$$H_1$$${H}_{1}$and$$H_2$$${H}_{2}$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$${S}^{d}$, which contains a great subsphere of dimension$$d-2$$$d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$${S}_{2,+}^{d}$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$$logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$${S}_{2,+}^{d}$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

more » « less