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Abstract s concave function on as a convex set in$${\mathbb {R}}^d$$ ${R}^{d}$ we derive a simple formula for the integral of its$${\mathbb {R}}^{d+1},$$ ${R}^{d+1},$s 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 logconcave function is logconcave as a function of the center of polarity. Also, we define the Santaló regions fors concave and logconcave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point. 
Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth_M ( , Mathematische Annalen)
Abstract Consider two halfspaces
and$$H_1^+$$ ${H}_{1}^{+}$ in$$H_2^+$$ ${H}_{2}^{+}$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ ${R}^{d+1}$ and$$H_1$$ ${H}_{1}$ are orthogonal and pass through the origin. The intersection$$H_2$$ ${H}_{2}$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ ${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$d dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ ${S}^{d}$ and is called a spherical wedge. Choose$$d2$$ $d2$n independent random points uniformly at random on 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$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ $logn$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a halfsphere.$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$