The Brunn–Minkowski Inequality, Minkowski's First Inequality, and Their Duals
Copyright policy )The Brunn–Minkowski Inequality, Minkowski's First Inequality, and Their Duals
Let \(K,L\) be convex bodies in Euclidean space \(\mathbb{E}^n\) with volumes \(V(K)=V(L)=1\), and let \(V_1(K,L)\) denote the mixed volume \(V(K, \dots, K,L)\). Then \[ V(K+L)^{1/n} -2\leq V_1(K,L) -1\leq {1\over n}\bigl(V(K+L)-2^n \bigr). \] These inequalities provide a quantitative improvement of the known equivalence of the Brunn-Minkowski inequality and Minkowski's first inequality for convex bodies. The left-hand inequality is due to \textit{H. Groemer} [Geom. Dedicata 33, No.~1, 117-122 (1990; Zbl 0692.52005)], the right-hand one is proved here, together with a variant and analogues for star bodies in Lutwak's dual Brunn-Minkowski theory. The main results are shown to be best possible in a precise sense.
- Eastern Washington University United States
- Catholic University of the Sacred Heart Italy
Applied Mathematics, star body, dual Brunn-Minkowski theory, Minkowski's first inequality, dual Brunn–Minkowski inequality, Convex sets in \(n\) dimensions (including convex hypersurfaces), Brunn-Minkowski inequality, convex body, Brunn–Minkowski inequality, dual mixed volume, Convex body, Inequalities and extremum problems involving convexity in convex geometry, geometric tomography, Brunn-Minkowski theorem, Analysis, dual Brunn-Minkowski inequality, mixed volume
Applied Mathematics, star body, dual Brunn-Minkowski theory, Minkowski's first inequality, dual Brunn–Minkowski inequality, Convex sets in \(n\) dimensions (including convex hypersurfaces), Brunn-Minkowski inequality, convex body, Brunn–Minkowski inequality, dual mixed volume, Convex body, Inequalities and extremum problems involving convexity in convex geometry, geometric tomography, Brunn-Minkowski theorem, Analysis, dual Brunn-Minkowski inequality, mixed volume
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