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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)

MSC:
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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