On differential geometry in the large. I. Minkowski’s problem
Hans Lewy · Transactions of the American Mathematical Society · 1938
Introduction.Hermann Minkowski, f in a fundamental paper on convex bodies, proposed the following Problem (M) : to determine a convex, threedimensional body B whose surface admits of a given Gaussian curvature K(n) >0, assigned as a continuous function of the direction of the interior normal n to the surface.After having stated three obviously necessary conditions for the function K(n), Minkowski proceeds to solve an analogous Problem (M') for polyhedra.He then considers a passage to the limit among the solutions of problems (M7) approximating (M), and establishes their convergence to a convex body B0.This construction leaves open the question as to whether B0 is a solution of (M).Minkowski remarks that B, if it exists, is, to within a translation, the uniquely determined solution of a certain third Problem (M") of the calculus of variations and that B0, too, is a solution of (M").Now if we assume (H) : the surface of B0 is differentiable to a sufficiently high order, then B0 solves (M).However, Minkowski does not discuss (H), but proves instead that the mixed volume V(B0, B0, C) of B0 with an arbitrary convex body C may be computed as though B0 were a solution of (M) and that, furthermore, a convex body is, except for a translation, uniquely determined by its mixed volume with the totality of convex bodies.While later authors have modified Minkowski's methods, there has been no improvement of his results as far as the hypothesis (H) is concerned.Thus Minkowski's results are open to the same criticism that could be raised against the early solutions of Plateau's problem: namely, that instead of solving the proposed problem, a more general problem is treated whose solution coincides with that of the former only if the latter solution satisfies certain highly restrictive conditions, and no indication is presented that these conditions are actually satisfied.The present paper contains a solution of (M) for the case of analytic K(n).It does not involve the Brunn-Minkowski inequalities, nor, indeed, the idea of mixed volume.It uses instead the author's results on elliptic and * Presented to the Society,