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Gabriele Bianchi

Publications and source records attributed to Gabriele Bianchi.

At least 19 recordsLinked to original sources
arXiv2025

Dark Coriolis Fields

We argue that the standard post-Newtonian expansion scheme used in General Relativity leaves room for time-space components $g_{ti}$ of the metric to be of the same order of the usual gravitational potential. We explore this possibility and find that such leading order contributions to $g_{ti}$ are related to the Coriolis field of Newton-Cartan gravity. We investigate the possibility that Coriolis fields mimic dark matter effects in disk galaxies. We find solutions from their field equations that sustain the velocity rotation curves in the bulge region and beyond it, notably describing flattish velocity profiles. We dub such solutions Dark Coriolis Fields.

gr-qc↗
arXiv2025

Approximation of rearrangements by polarizations

The symmetric decreasing rearrangement of functions on $\mathbb{R}^n$ features in several seminal inequalities, such as the P\'olya-Szeg\H{o} inequality. The latter was shown by the authors to hold for all smoothing rearrangements, a class that includes the more general $(k,n)$-Steiner rearrangement, as well as others introduced by Brock and by Solynin. The theory of rearrangements and their associated set maps is developed, with an emphasis on approximation, particularly by polarizations. The P\'olya-Szeg\H{o} inequality holds with equality for polarizations, so is proved relatively easily for rearrangements that can be suitably approximated by them. One goal here is to show that the Brock rearrangements cannot be approximated in such a way. It turns out that under mild conditions, each set map associated with a rearrangement has in turn an associated contraction map from $\mathbb{R}^n$ to $\mathbb{R}^n$. With this new analytical tool, several general results on the approximation of rearrangements are also proved.

math.FA↗
arXiv2025

Strict concavity properties of cross covariograms

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 n>1, and an analysis of how it can fail. Among the implications are that (i.) the cross covariogram of strictly convex bodies is strictly 1/n-concave, unless one body contains a translate of the other in its interior, and (ii.) the cross covariogram of an arbitrary convex body with its reflection through the origin is strictly log-concave.

math.MG↗
arXiv2024

Anisotropic symmetrization, convex bodies, and isoperimetric inequalities

This work is concerned with a P\'olya-Szeg\"o type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach that uncovers geometric aspects of the inequality is proposed. It relies upon anisotropic isoperimetric inequalities, fine properties of Sobolev functions, and results from the Brunn-Minkowski theory of convex bodies. Importantly, unlike previously available proofs, the one offered in this paper does not require approximation arguments and hence allows for a characterization of extremal functions.

math.FA↗
arXiv2022

The Pólya-Szegő inequality for smoothing rearrangements

A basic version of the Pólya-Szegő inequality states that if $Φ$ is a Young function, the $Φ$-Dirichlet energy -- the integral of $Φ(\|\nabla f\|)$ -- of a suitable function $f\in \mathcal{V}(\mathbb{R}^n)$, the class of nonnegative measurable functions on $\mathbb{R}^n$ that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions $f\in \mathcal{V}(\mathbb{R}^n)$, functions $f\in W^{1,p}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$ (when $1\le p<\infty$ and $Φ(t)=t^p$), and functions $f\in W^{1,1}_{loc}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$. In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the Pólya-Szegő inequality previously available under a common and very general framework.

math.FA↗
arXiv2022

Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces

Two related questions are discussed. The first is when reflection symmetry in a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-1\}$, implies full rotational symmetry, i.e., the closure of the group generated by the reflections equals $O(n)$. For $i=n-1$, this has essentially been solved by Burchard, Chambers, and Dranovski, but new results are obtained for $i\in \{1,\dots,n-2\}$. The second question, to which an essentially complete answer is given, is when (full) rotational symmetry with respect to a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-2\}$, implies full rotational symmetry, i.e., the closure of the group generated by all the rotations about each of the subspaces equals $SO(n)$. The latter result also shows that a closed set in $\mathbb{R}^n$ that is invariant under rotations about more than one axis must be a union of spheres with their centers at the origin.

math.MG↗
arXiv2022

Convergence of symmetrization processes

Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of $i$-dimensional subspaces of $\mathbb{R}^n$. Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions $i$ of the subspaces, are found, by combining two groups of results. The first, published separately, provides finite sets ${\mathcal{F}}$ of subspaces such that reflection symmetry (or rotational symmetry) with respect to each subspace in ${\mathcal{F}}$ implies full rotational symmetry. In the second, proved here, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set ${\mathcal{F}}$ of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in ${\mathcal{F}}$ appearing infinitely often in the sequence. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of $i$-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain's theorem is shown to hold for Schwarz symmetrization of compact sets.

math.MG↗
arXiv2020

Rearrangement and polarization

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 $\mathbb{R}^n$. Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally.

math.MG↗
arXiv2018

The $L_p$-Minkowski problem for $-n < p< 1$

Chou and Wang's existence result for the $L_p$-Minkowski problem on ${\mathbb S}^{n-1}$ for $p\in(-n,1)$ and an absolutely continuous measure $μ$ is discussed and extended to more general measures. In particular, we provide an almost optimal sufficient condition for the case $p\in(0,1)$.

math.CA↗
arXiv2016

The covariogram and Fourier-Laplace transform in $\mathbb{C}^n$

The covariogram $g_{K}$ of a convex body $K$ in $\mathbb{R}^n$ is the function which associates to each $x\in\mathbb{R}^n$ the volume of the intersection of $K$ with $K+x$. Determining $K$ from the knowledge of $g_K$ is known as the Covariogram Problem. It is equivalent to determining the characteristic function $1_K$ of $K$ from the modulus of its Fourier transform $\hat{1_K}$ in $\mathbb{R}^n$, a particular instance of the Phase Retrieval Problem. We connect the Covariogram Problem to two aspects of the Fourier transform $\hat{1_K}$ seen as a function in $\mathbb{C}^n$. The first connection is with the problem of determining $K$ from the knowledge of the zero set of $\hat{1_K}$ in $\mathbb{C}^n$. To attack this problem T. Kobayashi studied the asymptotic behavior at infinity of this zero set. We obtain this asymptotic behavior assuming less regularity on $K$ and we use this result as an essential ingredient for proving that when $K$ is sufficiently smooth and in any dimension $n$, $K$ is determined by $g_K$ in the class of sufficiently smooth bodies. The second connection is with the irreducibility of the entire function $\hat{1_K}$. This connection also shows a link between the Covariogram Problem and the Pompeiu Problem in integral geometry.

math.MG↗
arXiv2014

Covariograms generated by valuations

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 the function associating to each x \in \mathbb{R}^n the value ϕ(K \cap (K+x)). If ϕis the volume, then g_{K,ϕ} is the covariogram, extensively studied in various sources. When ϕis a quermassintegral (e.g., surface area or mean width) g_{K,ϕ} has been introduced by Nagel. We study various properties of ϕ-covariograms, mostly in the case n=2 and under the assumption that ϕis translation invariant, monotone and even. We also consider the generalization of Matheron's covariogram problem to the case of ϕ-covariograms, that is, the problem of determining an unknown convex body K, up to translations and point reflections, by the knowledge of g_{K,ϕ}. A positive solution to this problem is provided under different assumptions, including the case that K is a polygon and ϕis either strictly monotone or ϕis the width in a given direction. We prove that there are examples in every dimension n\geq3 where K is determined by its covariogram but it is not determined by its width-covariogram. We also present some consequence of this study in stochastic geometry.

math.MG↗
arXiv2014

The Blaschke-Santalo Inequality

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 paper we give a Fourier analytic proof of this fact.

math.MG↗
arXiv2012

Convergence in shape of Steiner symmetrizations

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 examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S^{n-1}.) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.

math.MG↗
arXiv2010

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

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 its Fourier transform \hat{1_K}. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k^n) noisy covariogram measurements to construct a convex polytope P_k that approximates K or its reflection -K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between P_k and K or -K tends to zero as k tends to infinity.

math.MG↗
arXiv2010

Covariogram of non-convex sets

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 covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than 9 points that have equal discrete covariogram.

math.MG↗