Erwin Lutwak - The Minkowski problem after 100 years--lots of new questions

The celebrated Minkowski problem (now one hundred years old) has had a significant impact on Twentieth Century mathematics, from discrete and convex geometry to partial differential equations.

The original formulation of the Minkowski problem asks for necessary and sufficient conditions on a given set of vectors in order that there exist a polytope each of whose faces has one of the given vectors as an outer normal and such that the area of each face of the polytope is equal to the length of its corresponding normal vector. In his review of Pogorelov's book on the Minkowski problem, Calabi stated ``The importance of the Minkowski problem and its solution is to be felt both in differential geometry and elliptic partial differential equations, on either count going far beyond the impact that the literal statement superficially may have. From the geometric view point it is the Rosetta Stone, from which several related problems can be solved.''

It has recently come to be recognized that the Minkowski problem is part of a family of related problems. These problems appear naturally in the study of abrasion of metals, in affine differential geometry, as well as in geometric convexity. Special cases of the one-dimensional versions of these problems can be reformulated as a family of questions regarding polygons in the Euclidean plane. Some of these questions can be explained to, appreciated by, and probably even answered by talented high school students. The higher-dimensional versions of these problems lead to deep questions about a fascinating class of elliptic partial differential equations.