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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)
##### Keywords:
extended affine surface area
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