In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski.
Define a quantity V1(K, L) by where V denotes the n-dimensional Lebesgue measure and + denotes the Minkowski sum.
One can show that the Brunn–Minkowski inequality for convex bodies in Rn implies Minkowski's first inequality for convex bodies in Rn, and that equality in the Brunn–Minkowski inequality implies equality in Minkowski's first 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 Rn: if K is a convex body in Rn, then with equality if and only if K is a ball of some radius.
This hyperbolic geometry-related article is a stub.