66 citations · 218 across the 20 of their papers we have counts for
16 papers · 1 filter
Strict concavity properties of cross covariograms
Gabriele Bianchi, Almut Burchard, Lawrence Lin
It is well-known that the cross covariogram of two convex bodies in n dimensions is 1/n-concave on its support. This paper provides conditions for strict 1/n-concavity in dimension…
Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces
Gabriele Bianchi, Richard J. Gardner, Paolo Gronchi
Two related questions are discussed. The first is when reflection symmetry in a finite set of -dimensional subspaces, , implies full rotational symmetry, i…
Rearrangement and polarization
Gabriele Bianchi, Richard J. Gardner, Paolo Gronchi +1
The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on . Rearrangements are map…
Convergence of symmetrization processes
Gabriele Bianchi, Richard J. Gardner, Paolo Gronchi
Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are inves…
Smoothness in the Minkowski problem for
Gabriele Bianchi, Károly J. Böröczky, Andrea Colesanti
We discuss the smoothness and strict convexity of the solution of the Minkowski problem when and the given measure has a positive density function.
The covariogram and Fourier-Laplace transform in
Gabriele Bianchi
The covariogram of a convex body in is the function which associates to each the volume of the intersection of with . Determi…
The Blaschke-Santalo Inequality
Gabriele Bianchi, Michael Kelly
The Blaschke-Santaló Inequality is the assertion that the volume product of a centrally symmetric convex body in Euclidean space is maximized by (and only by) ellipsoids. In this p…
Covariograms generated by valuations
Gennadiy Averkov, Gabriele Bianchi
Let ϕbe a real-valued valuation on the family of compact convex subsets of \mathbb{R}^n and let K be a convex body in \mathbb{R}^n. We introduce the ϕ-covariogram g_{K,ϕ} of K as t…
Convergence in shape of Steiner symmetrizations
Gabriele Bianchi, Almut Burchard, Paolo Gronchi +1
There are sequences of directions such that, given any compact set K in R^n, the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examp…
A countable set of directions is sufficient for Steiner symmetrization
Gabriele Bianchi, Daniel A. Klain, Erwin Lutwak +2
A countable dense set of directions is sufficient for Steiner symmetrization, but the order of directions matters.
Covariogram of non-convex sets
Carlo Benassi, Gabriele Bianchi, Giuliana D'Ercole
The covariogram of a compact set A contained in R^n is the function that to each x in R^n associates the volume of A intersected with (A+x). Recently it has been proved that the co…
Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms
Gabriele Bianchi, Richard J. Gardner, Markus Kiderlen
We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of i…
The cross covariogram of a pair of polygons determines both polygons, with a few exceptions
Gabriele Bianchi
The cross covariogram g_{K,L} of two convex sets K and L in R^n is the function which associates to each x in R^n the volume of the intersection of K and L+x. Very recently Averkov…
The covariogram determines three-dimensional convex polytopes
Gabriele Bianchi
The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determin…
Confirmation of Matheron's conjecture on the covariogram of a planar convex body
Gennadiy Averkov, Gabriele Bianchi
The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that f…
Retrieving convex bodies from restricted covariogram functions
Gennadiy Averkov, Gabriele Bianchi
The covariogram g_K(x) of a convex body K \subseteq E^d is the function which associates to each x \in E^d the volume of the intersection of K with K+x. Matheron asked whether g_K…