Titel: Retrieving Convex Bodies from Restricted Covariogram Functions


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Abstract:
The covariogram $g_K(x)$ of a convex body $K \subseteq \mathcal{E}^d$
is the function which associates to each $x \in \mathcal{E}^d$ the volume of
the intersection of $K$ with $K+x.$ Matheron (1986)
asked whether $g_K$ determines $K$, up to translations
and reflections in a point. Positive answers to Matheron's
question have been obtained for large classes of planar convex
bodies, while for $d\geq3$ there are both positive and negative results.

We sharpen some of the known
results on Matheron's conjecture indicating how much of the
covariogram information is needed to get the uniqueness of
determination.
We indicate some subsets of the support of the covariogram, with
arbitrarily small Lebesgue measure, such that the covariogram,
restricted to those subset, identifies certain geometric properties of
the body.
These results are more precise in the planar case, but some of them,
both positive and negative ones, are proved for bodies of any
dimension.
Moreover some results regard most convex bodies, in the Baire category sense.
Another group of results is concerned with extending the class of convex
bodies for which Matheron's conjecture is confirmed.