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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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