Generalized translation invariant valuations and the polytope algebra

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.Comment: 29 pages; minor change

On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus

Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n$, we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant function. The $k$-dimensional subtorus $M$ is chosen randomly and uniformly. We show that when $k \leq c \log n / (\log \log n + \log 1/\varepsilon)$, the maximum and the minimum of $f$ on this random subtorus $M$ differ by at most $\varepsilon$, with high probability.Comment: 8 page

The Fourier transform on valuations is the Fourier transform

Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant smooth valuations that has the same functorial properties as the classical Fourier transform. In this paper, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker--Fourier transform. One of these properties was previously only conjectured by Alesker and is proved here for the first time.Comment: 32 page

Convex valuations, from Whitney to Nash

We consider the Whitney problem for valuations: does a smooth $j$-homogeneous translation-invariant valuation on $\mathbb R^n$ exist that has given restrictions to a fixed family $S$ of linear subspaces? A necessary condition is compatibility: the given valuations must coincide on intersections. We show that for $S=\mathrm{Gr}_r(\mathbb R^n)$, the grassmannian of $r$-planes, this condition becomes sufficient once $r\geq j+2$. This complements the Klain and Schneider uniqueness theorems with an existence statement, and provides a recursive description of the image of the cosine transform. Informally speaking, we show that the transition from densities to valuations is localized to codimension $2$. We then look for conditions on $S$ when compatibility is also sufficient for extensibility, in two distinct regimes: finite arrangements of subspaces, and compact submanifolds of the grassmannian. In both regimes we find unexpected flexibility. As a consequence of the submanifold regime, we prove a Nash-type theorem for valuations on compact manifolds, from which in turn we deduce the existence of Crofton formulas for all smooth valuations on manifolds. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.Comment: 53 page