Minkowski's first inequality for convex bodies
In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality. Statement of the inequality. Let "K" and "L" be two "n"-dimensional convex bodies in "n"-dimensional Euclidean space R. Define a quantity "V"("K", "L") by where "V" denotes the "n"-dimensional Lebesgue measure and + denotes the Minkowski sum. Then with equality if and only if "K" and "L" are homothetic, i.e. are equal up to translation and dilation. Connection to  other inequalities. The Brunn–Minkowski inequality. One can show that the Brunn–Minkowski inequality for convex bodies in R implies Minkowski's first inequality for convex bodies in R, and that equality in the Brunn–Minkowski inequality implies equality in Minkowski's first inequality. The isoperimetric inequality. By taking "L" = "B", the "n"-dimensional unit ball, in Minkowski's first inequality for convex bodies, one obtains the isoperimetric inequality for convex bodies in R: if "K" is a convex body in R, then with equality if and only if "K" is a ball of some radius. Minkowski's first inequality