The Brunn-Minkowski inequality, a fundamental result in geometry and analysis, states that for convex bodies \(K\) and \(L\) in \(\mathbb{R}^n\), the volume of their Minkowski sum \((1 - \lambda)K + \lambda L\) satisfies: \[ V((1 - \lambda)K + \lambda L)^{1/n} \geq (1 - \lambda)V(K)^{1/n} + \lambda V(L)^{1/n}, \] with equality if and only if \(K\) and \(L\) are homothetic. This inequality is a cornerstone of the Brunn-Minkowski theory, which has applications in various areas of mathematics and physics. The article by R. J. Gardner provides an overview of the Brunn-Minkowski inequality, its historical context, and its connections to other inequalities and applications. Key points include: - The Brunn-Minkowski inequality implies the classical isoperimetric inequality for important classes of subsets in \(\mathbb{R}^n\). - It has been extended to more general spaces and has applications in analysis, probability theory, and statistics. - The inequality is related to other inequalities such as Minkowski's first inequality, the Sobolev inequality, and the Prékopa-Leindler inequality. - The Brunn-Minkowski inequality is used to prove the Wulff shape of crystals and to study surface area measures. - It appears in the work of McCann on interacting gases and transport of mass, where it is used to derive new global convexity inequalities. The article also discusses the proof of the Brunn-Minkowski inequality, the equality conditions, and its implications in various mathematical contexts.The Brunn-Minkowski inequality, a fundamental result in geometry and analysis, states that for convex bodies \(K\) and \(L\) in \(\mathbb{R}^n\), the volume of their Minkowski sum \((1 - \lambda)K + \lambda L\) satisfies: \[ V((1 - \lambda)K + \lambda L)^{1/n} \geq (1 - \lambda)V(K)^{1/n} + \lambda V(L)^{1/n}, \] with equality if and only if \(K\) and \(L\) are homothetic. This inequality is a cornerstone of the Brunn-Minkowski theory, which has applications in various areas of mathematics and physics. The article by R. J. Gardner provides an overview of the Brunn-Minkowski inequality, its historical context, and its connections to other inequalities and applications. Key points include: - The Brunn-Minkowski inequality implies the classical isoperimetric inequality for important classes of subsets in \(\mathbb{R}^n\). - It has been extended to more general spaces and has applications in analysis, probability theory, and statistics. - The inequality is related to other inequalities such as Minkowski's first inequality, the Sobolev inequality, and the Prékopa-Leindler inequality. - The Brunn-Minkowski inequality is used to prove the Wulff shape of crystals and to study surface area measures. - It appears in the work of McCann on interacting gases and transport of mass, where it is used to derive new global convexity inequalities. The article also discusses the proof of the Brunn-Minkowski inequality, the equality conditions, and its implications in various mathematical contexts.