A Theorem on Convex Bodies of the Brunn-Minkowski Type.

The Brunn-Minkowski-Type Inequality

Volume Inequalities and Additive Maps of Convex Bodies

Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Mi...

Fourier Transform and Firey Projections of Convex Bodies

In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, what he called p-sums. E. Lutwak [Lu2], [Lu3] showed that these Firey sums lead to a Brunn-Minkowski theory for each p ≥ 1. He introduced the notions of p-mixed volume, p-surface area measure, and proved an integral representation and inequalities for p−mixed volume...

The Fourier Transform and Firey Projections of Convex Bodies

In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, that he called p-sums. Lutwak [Lu2], [Lu3] showed that these Firey sums lead to a Brunn-Minkowski theory for each p ≥ 1. He introduced the notions of p-mixed volume, p-surface area measure, and proved an integral representation and inequalities for p-mixed volumes, ...

The Matrix Analogs of Firey’s Extension of Minkowski Inequality and of Firey’s Extension of Brunn-Minkowski Inequality

– The Brunn-Minkowski theory is a central part of convex geometry. At its foundation lies the Minkowski addition of convex bodies which led to the definition of mixed volume of convex bodies and to various notions and inequalities in convex geometry. Its origins were in Minkowski’s joining his notion of mixed volumes with the Brunn-Minkowski inequality, which dated back to 1887. Since then it h...

On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality μ(λA+ (1− λ)B) ≥ λμ(A) + (1− λ)μ(B) holds true for an unconditional product measure μ with decreasing density and a pair of unconditional convex bodies A,B ⊂ R. We also show that the above inequality is true for any unconditional logconcave me...