Converse theorem for the Minkowski inequality

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...

the brunn-minkowski-type inequality

The Hasse–Minkowski Theorem

The Hasse-Minkowski Theorem provides a characterization of the rational quadratic forms. What follows is a proof of the Hasse-Minkowski Theorem paraphrased from the book, Number Theory by Z.I. Borevich and I.R. Shafarevich [1]. Throughout this paper, some familiarity with the p-adic numbers and the Hilbert symbol is assumed and some basic facts about quadratic forms are stated without proof. Al...

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.