Minkowski inequality for nearly spherical domains

The Brunn-Minkowski-Type Inequality

The Brunn-Minkowski Inequality

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...

Brunn - Minkowski Inequality

– We present a one-dimensional version of the functional form of the geometric Brunn-Minkowski inequality in free (noncommutative) probability theory. The proof relies on matrix approximation as used recently by P. Biane and F. Hiai, D. Petz and Y. Ueda to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered ...

Quantitative stability for the Brunn-Minkowski inequality

We prove a quantitative stability result for the Brunn-Minkowski inequality: if |A| = |B| = 1, t ∈ [τ, 1−τ ] with τ > 0, and |tA+(1−t)B| ≤ 1+δ for some small δ, then, up to a translation, both A and B are quantitatively close (in terms of δ) to a convex set K.

the brunn-minkowski-type inequality