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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.

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
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