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- W2044672008 abstract "Various means (over the unit sphere) of powers of the curvature function of a convex body arise in geometric convexity. For example, both the surface area and the affine surface area of sufficiently smooth bodies can be expressed as such means. The survey of Gruber [ 111 on approximation of convex bodies contains a number of examples of means of powers of the curvature function arising in approximation problems (see, pp. 137 141. 143, and 1466148 of [ 11 I). These means arise, surprisingly, in questions related to approximating convex bodies by polytopes (see, e.g., McClure and Vitale [18], Schneider [24], and Schneider and Wieacker [27]). An interesting conjecture of Firey [9, p. S] (also [26, p. 2571). which was recently proved by Gage [lo] for plane convex bodies, is that a certain weighted mean of the curvature function lies between the arithmetic and harmonic means of the curvature function. It turns out that there is also a close connection between various means of the curvature function of a convex body and some mixed volumes involving the body. In this note we investigate means of the curvature function of a convex body which arise as coefficients in the expansion of the afftne surface area of outer and inner Blaschke parallel bodies. For a variety of reasons we will take a general approach to the study of these means, an approach similar to that taken in [14] (see also [4, pp. 164.-1671) in presenting dual mixed volumes. Let .X” denote the set of convex bodies (compact convex sets with interior points) in Euclidean n-space R”. We use h( K, .) to denote the support function of a convex body K; i.e.." @default.
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- W2044672008 date "1987-08-01" @default.
- W2044672008 modified "2023-10-13" @default.
- W2044672008 title "Mixed affine surface area" @default.
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- W2044672008 doi "https://doi.org/10.1016/0022-247x(87)90097-7" @default.
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