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The convex floating body. (English) Zbl 0739.52008
There are several ways to generalize the classical concept of affine surface area of a sufficiently smooth convex body \(K\) in \(\mathbb{R}^ n\) due to Blaschke to arbitrary convex bodies [see K. Leichtweiss, Manuscr. Math. 56, 429-464 (1986; Zbl 0588.52011), E. Lutwak, Adv. Math. 85, No. 1, 39-68 (1991; Zbl 0727.53016) and C. M. Petty, Ann. N. Y. Acad. Sci. 440, 113-127 (1985; Zbl 0576.52003)].
For \(\delta>0\) the convex floating body \(K_ \delta\) of \(K\) is the intersection of all halfspaces the complements of which intersect \(K\) in a set of volume \(\delta\). For \(x\in\partial K\) let \(\Delta(x,\delta)\) denote the height of a slice of volume \(\delta\) cut off from \(K\) by a hyperplane normal to the normal of \(K\) at \(x\). W. Blaschke [Vorlesungen über Differentialgeometrie. II., Berlin (1923)] \((n=3)\) and K. Leichtweiss [Stud. Sci. Math. Hungar. 21, 453-474 (1986; Zbl 0561.53012)] (general \(n\)) proved that for sufficiently smooth \(K\) the ordinary affine surface area of \(\partial K\) equals \(\lim_{c_ n}(V(K)-V(K_ \delta))/\delta^{2/(n+1)}\) as \(\delta\to 0\) where \(c_ n\) is a suitable constant. The authors show that for any convex body \(K\) the following equality holds as \(\delta\to 0\): \[ \lim_{c_ n}(V(K)- V(K_ \delta))/\delta^{2/(d+1)}=\int(\lim_{c_ n}\Delta(x,\delta)/ ( \delta^{2/(n+1)})d\mu(x) \] where \(\mu\) is the surface area measure on \(\partial K\) and the integral is extended over \(\partial K\). Hence the integral may be used as a further definition of the concept of affine surface area of an arbitrary convex body.
Reviewer: P.Gruber (Wien)

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
53A15 Affine differential geometry
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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