The Brunn-Minkowski inequality is a fundamental result in geometry and analysis, closely related to the isoperimetric inequality. It states that for convex bodies K and L in $ \mathbb{R}^n $, and $ 0 < \lambda < 1 $, the inequality $ V((1-\lambda)K + \lambda L)^{1/n} \geq (1-\lambda)V(K)^{1/n} + \lambda V(L)^{1/n} $ holds, where V denotes volume. This inequality is easy to prove and quickly implies the classical isoperimetric inequality for important classes of sets in $ \mathbb{R}^n $. The Brunn-Minkowski inequality has been extended to various settings, including Lebesgue measurable sets and has significant applications in analysis, probability theory, and geometry. It is also related to other important inequalities such as the Prékopa-Leindler inequality, which is a reverse form of Hölder's inequality. The Brunn-Minkowski inequality has been used to derive various results in convex geometry, including the isoperimetric inequality, the Sobolev inequality, and the entropy power inequality. It has also been applied to problems in physics, such as the Wulff shape of crystals and the study of gases. The inequality has a wide range of applications and is a central topic in the study of convex geometry and analysis.The Brunn-Minkowski inequality is a fundamental result in geometry and analysis, closely related to the isoperimetric inequality. It states that for convex bodies K and L in $ \mathbb{R}^n $, and $ 0 < \lambda < 1 $, the inequality $ V((1-\lambda)K + \lambda L)^{1/n} \geq (1-\lambda)V(K)^{1/n} + \lambda V(L)^{1/n} $ holds, where V denotes volume. This inequality is easy to prove and quickly implies the classical isoperimetric inequality for important classes of sets in $ \mathbb{R}^n $. The Brunn-Minkowski inequality has been extended to various settings, including Lebesgue measurable sets and has significant applications in analysis, probability theory, and geometry. It is also related to other important inequalities such as the Prékopa-Leindler inequality, which is a reverse form of Hölder's inequality. The Brunn-Minkowski inequality has been used to derive various results in convex geometry, including the isoperimetric inequality, the Sobolev inequality, and the entropy power inequality. It has also been applied to problems in physics, such as the Wulff shape of crystals and the study of gases. The inequality has a wide range of applications and is a central topic in the study of convex geometry and analysis.