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Creators/Authors contains: "Werner, Elisabeth_M"

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  1. Abstract

    Using a natural representation of a 1/s-concave function on$${\mathbb {R}}^d$$Rdas a convex set in$${\mathbb {R}}^{d+1},$$Rd+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.

     
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  2. Abstract

    Consider two half-spaces$$H_1^+$$H1+and$$H_2^+$$H2+in$${\mathbb {R}}^{d+1}$$Rd+1whose bounding hyperplanes$$H_1$$H1and$$H_2$$H2are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$S2,+d:=SdH1+H2+is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$Sd, which contains a great subsphere of dimension$$d-2$$d-2and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$S2,+dand 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$${\mathbb {S}}_{2,+}^d$$S2,+d. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

     
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