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.
scientific article; zbMATH DE number 1471627