| Citation | Preiss is a real analyst of great distinction. In his early work he answered several fundamental problems of classical real analysis. For example he obtained a very surprising characterization of real-valued functions of the first Baire class in terms of three everywhere differentiable functions, and answered a problem of Klee by proving, using descriptive set theory, that every convex Borel set in a finite-dimensional space can be obtained from convex open sets by operations preserving both convexity and the Borel structure. He is well-known for deep results in geometric measure theory, for example, that every measure in Rn with finite and positive m-dimensional density almost everywhere is concentrated on a countable union of m-dimensional C1 submanifolds. Seminal contributions to nonlinear geometric functional analysis include his proof that real-valued Lipschitz functions on Banach spaces with separable dual are somewhere differentiable. In recent work he has found with Kirchheim spectacular counter-examples of Lipschitz mappings from R2 to R2 whose gradients take finitely many values Ai almost everywhere with no rank-one differences between the Ai. |