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The distribution function of the convolution square of a convex symmetric body in \({\mathbb{R}{}}^ n\). (English) Zbl 0774.52004
Let \(C\) be a centrally symmetric convex body in \(\mathbb{R}^ n\) with volume 1 and put \[ \begin{aligned} C(\delta): & = \{x\in\mathbb{R}^ n:\text{Vol} C\cap(C+x)\geq\delta\}\quad\text{ and } \\ V_ C(\delta): & = \text{ Vol } \{x\in\mathbb{R}^ n:\text{Vol} C\cap(C+x)>\delta\} \end{aligned} \] for \(0\leq\delta\leq 1\). The author relates these notions to some familiar objects in the affine geometry of convex bodies. Typical results are the limit relations \[ \lim_{\delta\to 1}(1-\delta)^{-1}C(\delta)=P^* \] in the Hausdorff metric, where \(P^*\) is the polar of the projection body of \(K\), and \[ \lim_{\delta\to 0}{V_ C(0)-V_ C(\delta)\over\delta^{2/(n+1)}}=a_ nS_{\text{aff}}(C), \] where \(a_ n\) is an explicit constant and \(S_{\text{aff}}\) denotes the (generalized) affine surface area. Finally, the exact asymptotic behaviour of \(V_ C(0)-V_ C(\delta)\) for \(\delta\to 0\) is determined if \(C\) is a symmetric polytope. This is in analogy to work of C. Schütt [Isr. J. Math. 73, No. 1, 65-77 (1991; Zbl 0745.52006)], who used the convex floating body.

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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