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The Santaló-regions of a convex body. (English) Zbl 0917.52004
Let $$K$$ be a compact convex body in the space $$\mathbb{R}^n$$ with nonempty interior $$\text{int}(K)$$ and let $$K^x$$ be the polar body of $$K$$ with respect to a point $$x\in\text{int}(K)$$. Then the Blaschke-Santaló inequality says for the volume product of $$K$$ and $$K^x$$ that $$\min_{x\in \text{int}(K)}|K|\;|K^x|\leq v^2_n$$ $$(v_n$$ volume element of the unit ball in $$\mathbb{R}^n)$$. Motivated by an improvement of M. Meyer and A. Pajor [Arch. Math. 55, No. 1, 82-93 (1990; Zbl 0718.52011)] of this inequality, considering restricted domains for the volume product, the authors introduce, following E. Lutwak, the definition of the (strictly convex and affinely covariant) “Santaló region” $$S(K,t)$$ of $$K$$ to the parameter $$t\in\mathbb{R}^+$$ given by $S(K,t):=\left\{x\in\text{int}(K);{|K|\;|K^x|\over v^2_n}\leq t\right\}.$ They systematically investigate $$S(K,t)$$ and its relation to suitable dilates of $$K$$ (Theorem 9) resp. to suitable convex floating bodies of $$K$$ (Propos. 1 (v) and Propos. 14). One main result of the authors (Theorem 10) is the formula ${1\over 2}\left({|K|\over v_n}\right)^{2\over n+1} \text{ as }(K)=\lim_{t\to+\infty} {|K|-|S(K,t)|\over t^{-{2\over n+1}}}$ for the affine surface area as $$(K)$$ of $$K$$ which also may be used as a new definition of as $$(K)$$ coinciding with the previous ones.

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52A38 Length, area, volume and convex sets (aspects of convex geometry)
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