Approximation of continuous valuations by quasi-smooth valuations

Determine whether every continuous valuation on the space of convex bodies in Euclidean space can be approximated, in a suitable sense, by quasi-smooth valuations.

Background

The paper discusses Alesker’s notion of quasi-smooth valuations on the space of convex bodies in Euclidean space. These valuations satisfy a strong uniform differentiability condition for the scaling maps t ↦ μ(tK+x) at t=0, which permits their derivatives to be studied as translation-invariant valuations on tangent spaces.

The authors explicitly note that quasi-smooth valuations form a restricted class and that it is unknown whether arbitrary continuous valuations admit suitable approximation by valuations in this class. This question motivates the paper’s use of affine-smooth valuations on polytopes, for which a weaker differentiability property can be established directly.