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Inequalities for dual affine quermassintegrals. (English) Zbl 1094.52004
Let \(K,L\) be two star bodies in the \(n\)-dimensdional euclidean space containig the origin in their interiors; let \(\xi \in G(n,i)\) and \(0\leq p \leq i\); let \(\tilde V(K\cap \xi,i-p;L\cap \xi,p)\) denote the dual mixed volume where \(K\) appears \(i-p\) times and \(L\) appears \(p\) times. The authors introduce the notion of mixed \(p\)-dual affine quermassintegrals in the following way: \[ \tilde {\Phi}_{p,i}(K,L)=k_n \left( \int_{G(n,n-i)} \left[ \frac{\tilde V_{p,n-i}(K,L;\xi)}{k_{n-i}} \right]^n d\xi \right)^{\frac 1n} \] where \[ \tilde V_{p,i}(K,L;\xi)=\tilde V(K\cap \xi,i-p;L\cap \xi,p). \] They establish the Minkowski inequality for them: \[ \tilde {\Phi}_{p,i}(K,L)^{n-i}\leq \tilde {\Phi}_{p,i}(K,K)^{n-i-p} \tilde {\Phi}_{n-i,i}(K,L)^p, \] with equality if and only if \(K\) is a dilatate of \(L\). As an application, the dual Brunn-Minkowski inequality for the dual affine quermassintegrals is obtained. They also establish a connection between the affine quermassintegrals and the dual affine quermassintegrals for a given convex body.

52A39 Mixed volumes and related topics in convex geometry
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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[1] Bonnesen T, Fenchel W: Theorie der konvexen Körper. Springer, Berlin; 1934. · Zbl 0008.07708
[2] Federer H: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften. Volume 153. Springer, New York; 1969:xiv+676.
[3] Gardner RJ: Geometric Tomography, Encyclopedia of Mathematics and Its Applications. Volume 58. Cambridge University Press, Cambridge; 1995:xvi+424.
[4] Grinberg, EL, Isoperimetric inequalities and identities for[inlineequation not available: see fulltext.]-dimensional cross-sections of a convex bodies, London Mathematical Society, 22, 478-484, (1990) · Zbl 0721.52007
[5] Leichtweiss K: Konvexe Mengen. Springer, Berlin; 1980:330 pp. (loose errata).
[6] Lutwak, E, Dual mixed volumes, Pacific Journal of Mathematics, 58, 531-538, (1975) · Zbl 0273.52007
[7] Lutwak, E, A general isepiphanic inequality, Proceedings of the American Mathematical Society, 90, 415-421, (1984) · Zbl 0534.52011
[8] Lutwak, E, Inequalities for Hadwiger’s harmonic quermassintegrals, Mathematische Annalen, 280, 165-175, (1988) · Zbl 0617.52007
[9] Santaló LA: Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and Its Applications. Volume 1. Addison-Wesley, Massachusetts; 1976:xvii+404.
[10] Schneider R: Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications. Volume 44. Cambridge University Press, Cambridge; 1993:xiv+490.
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