arXiv subjects
Publications and source records attributed to Judit Abardia-Evéquoz.
We show the existence of additive kinematic formulas for general flag area measures, which generalizes a recent result by Wannerer. Building on previous work by the second named author, we introduce an algebraic framework to compute these formulas explicitly. This is carried out in detail in the case of the incomplete flag manifold consisting of all $(p+1)$-planes containing a unit vector.
A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0 \leq p \leq n-1$. Using local parallel sets, Hinderer constructed examples of $\mathrm{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\mathrm{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\mathrm{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger's theorem.
The space of continuous, ${\rm SL}(m,C)$-equivariant, $m\geq 2$, and translation covariant valuations taking values in the space of real symmetric tensors on $C^m\cong R^{2m}$ of rank $r\geq 0$ is completely described. The classification involves the moment tensor valuation for $r\geq 1$ and is analogous to the classification of the corresponding tensor valuations that are ${\rm SL}(2m,R)$-equivariant, although the method of proof cannot be adapted.
We provide a description of the space of continuous and translation invariant Minkowski valuations $Φ:\mathcal{K}^n\to\mathcal{K}^n$ for which there is an upper and a lower bound for the volume of $Φ(K)$ in terms of the volume of the convex body $K$ itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over $(n-1)$-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body.
We investigate Minkowski additive, continuous, and translation invariant operators $Φ:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex bodies such that the volume of the image $Φ(K)$ is bounded from above and below by multiples of the volume of the convex body $K$, uniformly in $K$. We obtain a representation result for an infinite subcone contained in the cone formed by this type of operators. Under the additional assumption of monotonicity or $SO(n)$-equivariance, we obtain new characterization results for the difference body operator.