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The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas. (English) Zbl 0853.52005

The aim of the article is to extend the known inequalities involving (extended) affine and geominimal surface areas to the so-called \(p\)-affine and \(p\)-geominimal surface areas for \(p >1\). All of these latter notions refer to the Firey linear combination \(\lambda \cdot K +_p \mu \cdot L\), given by \[ h(\lambda \cdot K +_p \mu \cdot L,\cdot) := (\lambda h(K,\cdot)^p + \mu h(L,\cdot)^p)^{1\over p} \] \((\lambda \geq 0\), \(\mu \geq 0\), not both zero), where \(h(K,u)\) resp. \(h(L,u)\) \((u \in S^{n-1})\) is the support function of a convex body \(K\) resp. \(L\) in euclidean \(n\)-space containing the origin \(o\) in its interior. From this a \(p\)-mixed volume \(V_p (K,L)\) may be derived by \[ {n\over p} V_p(K,L) := \lim_{\varepsilon \to 0} {V(K+_p \varepsilon \cdot L) - V(K) \over \varepsilon} = {1\over p} \int_{S^{n-1}} h(L,u)^p dS_p (K,u) \] \((S_p(K,\cdot) =\) suitable Borel measure on \(S^{n-1})\) and moreover
\(V_p(K,L^*) := {1\over n} \int_{S^{n-1}} \rho_L (u)^{-p} dS_p (K,u)\) for a star body \(L\) about \(o\) with the radial function \(\rho_L(u)\).
Now in continuation of part I of the author’s paper [part I: Mixed volumes and the Minkowski problem, J. Differ. Geom. 38, No. 1, 131-150 (1993; Zbl 0788.52007)] a \(p\)-affine surface area \(\Omega_p(K)\) is defined by \(\Omega_p(K)^{n+p \over n} = \inf_L nV_p (K,L^*) \cdot (nV(L))^{p \over n}\), in a certain sense generalizing a \(p\)-geominimal surface area \(G_p(K)\) with \(G_p(K) = \inf_{L'} nV_p (K,L') \cdot (\omega_n^{-1} V(L'{}^*))^{p\over n}\) where the infimum is only taken for the convex star bodies \(L'\) with polar bodies \(L'{}^*\) \((\omega_n = \) volume of the unit \(n\)-ball). As one of the big varieties of results the following \(p\)-extension of the affine isoperimetric inequality may be stated: \(\Omega_p(K)^{n+p} \leq n^{n+p} \omega^{2p}_n V(K)^{n-p}\) with equality if and only if \(K\) is an ellipsoid. This extension represents an improvement of the classical affine isoperimetric inequality because the \(p\)-th root of the \(p\)-affine isoperimetric ratio is shown to be monotone nondecreasing in \(p\). A similar result is proved for the \(p\)-geominimal surface area. As in the classical case there exists an inequality between \(p\)-affine and \(p\)-geominimal surface area.

MSC:

52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A39 Mixed volumes and related topics in convex geometry

Citations:

Zbl 0788.52007
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