In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, an...

In this paper, we first introduce a new concept of dual quermassintegral sum function of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov– Fenchel inequality and the Brunn–Minkowski inequality for mixed intersection...

In this paper we establish the Lp-Minkowski inequality and Lp-Aleksandrov-Fenchel type inequality for Lp-dual mixed volumes of star duality of mixed intersection bodies, respectively. As applications, we get some related results. The paper new contributions that illustrate this duality of projection and intersection bodies will be presented. M.S.C. 2000: 52A40.

In this paper, connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy conditions, and quantum inequality restrictions (uncertainty principle-type inequalities) on negative energy. In twoand four-dimensional Minkowski spacetime, we examine quantized, free massless, minimally-coupled scalar fields . In a two-dimensional spatially compactified Minkowski universe...

We present a simple proof of Christer Borell’s general inequality in the Brunn-Minkowski theory. We then discuss applications of Borell’s inequality to the log-Brunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang. 2010 Mathematics Subject Classification. Primary 28A75, 52A40.

We consider the second variation for the volume of convex bodies associated with the Lp Minkowski-Firey combination and obtain a Poincaré-type inequality on the Euclidean unit sphere Sn−1 . Mathematics subject classification (2010): 52A20.

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of R n. For the existence part we apply the variational method introduced by Jerison in: Adv. Math. 122 (1996), pp. 262–279. Uniqueness follows from the Brunn–Minkowski inequality for the torsional rigidity and corresponding ...

This article describes a new proof of the equality condition for the Brunn-Minkowski inequality. The Brunn-Minkowski Theorem asserts that, for compact convex sets K,L ⊆ Rn, the n-th root of the Euclidean volume Vn is concave with respect to Minkowski combinations; that is, for λ ∈ [0, 1], Vn((1− λ)K + λL) ≥ (1− λ)Vn(K) + λVn(L). The equality condition asserts that if K and L both have positive ...

In this paper, we establish an Orlicz dual of the log-Aleksandrov–Fenchel inequality, by introducing two new concepts mixed volume measures, and using newly established Aleksandrov–Fenchel inequality. The log-Aleksandrov– Fenchel inequality in special cases yields classical some logarithmic Minkowski type inequalities, respectively. Moreover, is therefore also derived.