Singular valuations and the Hadwiger theorem on convex functions

Abstract

We give a new proof of the Hadwiger theorem on convex functions derived from a characterization of smooth rotation invariant valuations. We also give a description of the construction of singular Hessian valuations using integration over the differential cycle and provide a new representation of these functionals as principal value integrals with respect to the Hessian measures.

Monge-Ampère operators and valuations

“…, f ) = µ(f ). As shown in [37], the polarization lifts to a distribution on (R n ) k . For complex-valued valuations, these distributions may be characterized in the following way.…”

“…It was shown in [37,Corollary 4.7] that the right hand side of this equation depends continuously on f ∈ Conv(R n , R). In particular, Ψ τ is weakly*-continuous.…”

“…To any such function f one can associate an integral current D(f ) on T * R n that is uniquely characterized by a list of natural properties satisfied by the graph of the differential of a C 2 -function. The current D(f ) is called the differential cycle of f and can be used to construct valuations on convex bodies [6] and convex functions [34,36,37].…”

From valuations on convex bodies to convex functions

“…Valuations on convex functions on R n have been one of the most active areas of study [3,7,10,11,12,13,14,15,16,17,25,26,27]. The primary focus is on subspaces of the space Conv(R n ) of all convex functions u : R n → (−∞, +∞] that are lower semicontinuous and proper, that is, not identically +∞.…”