arXiv subjects
Publications and source records attributed to Federico Glaudo.
We prove that any triple of complete flags in $\mathbb R^d$ admits a common generating set of size $\lfloor 5d/3\rfloor$ and that this bound is sharp. This result extends the classical linear-algebraic fact -- a consequence of the Bruhat decomposition of $\text{GL}_d(\mathbb R)$ -- that any pair of complete flags in $\mathbb R^d$ admits a common generating set of size $d$. We also deduce an analogue for $m$-tuples of flags with $m>3$.
For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for $2$-torsion-free groups $G$ by giving characterizations, both algebraic and combinatorial, of the fibers of $\text{FS}$. Equivalently, we characterize all pairs of multisets $A,B$ with $\text{FS}(A)=\text{FS}(B)$. Our results build on recent work of Ciprietti and the first author.
For every $d\ge 3$, we construct a noncompact smooth $d$-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below $1$. We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in $\mathbb R^d$. The examples we construct have nondegenerate asymptotic cone. The dimensional constraint $d\ge 3$ is sharp. Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed in nonnegatively curved spaces with nondegenerate asymptotic cones isoperimetric sets with large volumes always exist. This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets.
We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Tišer, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Tišer bound even for the corresponding statement in general metric spaces). More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.
The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^α|x-y|^{n+2s} |y|^α} \mathrm{d} x \, \mathrm{d} y \geq Λ_{n, s, p, α,β} \|u |x|^{-β}\|_{L^p}^2, $$ for $0<s<\min\{1, n/2\}$, $2<p<2^*_s$, and $α,β\in\mathbb R$ so that $β-α= s - n\big(\frac12 - \frac1p\big)$ and $-2s < α< \frac{n-2s}{2}$. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for $α\ge 0$. Furthermore, we show that minimizers remain symmetric when $α<0$ for $p$ very close to $2$. Our results fit into the more ambitious goal of understanding the symmetry region of the minimizers of the fractional Caffarelli-Kohn-Nirenberg inequality. We develop a general framework to deal with fractional inequalities in $\mathbb R^n$, striving to provide statements with a minimal set of assumptions. Along the way, we discover a Hardy-type inequality for a general class of radial weights that might be of independent interest.
We prove that a subset of the hypercube $(0,1)^d$ with volume sufficiently close to $\frac12$ has (relative) perimeter greater than or equal to $1$. This settles a conjecture by Brezis and Bruckstein. We also prove that, in contrast with what happens for the high-dimensional sphere $\mathbb S^d$, the isoperimetric profile of the hypercube $(0,1)^d$ does not converge to the Gaussian isoperimetric profile as $d\to\infty$.
Let $A$ be a multiset with elements in an abelian group. Let $FS(A)$ be the multiset containing the $2^{|A|}$ sums of all subsets of $A$. We study the reconstruction problem ``Given $FS(A)$, is it possible to identify $A$?'', and we give a satisfactory answer for all abelian groups. We prove that, up to identifying multisets through a natural equivalence relation, the function $A \mapsto FS(A)$ is injective (and thus the reconstruction problem is solvable) if and only if every order $n$ of a torsion element of the abelian group satisfies a certain number-theoretical property linked to the multiplicative group $(\mathbb{Z} / n\mathbb{Z})^*$. The core of the proof relies on a delicate study of the structure of cyclotomic units. Moreover, as a tool, we develop an inversion formula for a novel discrete Radon transform on finite abelian groups that might be of independent interest.
Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that $Lu=δ_0$. As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of $|x|$, plus homogeneous polynomials multiplied by $\log(|x|)$ if the dimension $n$ is even) which improves the classical result of F. John (1950). The control we have on the "complexity" of each homogeneous term is optimal and in particular, when $L$ is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949). The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by $\mathbb Z^2$.
We investigate the validity and the stability of various Minkowski-like inequalities for $C^1$-perturbations of the ball. Let $K\subseteq\mathbb R^n$ be a domain (possibly not convex and not mean-convex) which is $C^1$-close to a ball. We prove the sharp geometric inequality $$ \left(\int_{\partial K} \lVert II\rVert_1 d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_1(n)Per(K)^{\frac1{n-1}} , $$ where $C_1(n)$ is the constant that yields the equality when $K=B_1$ (and $\lVert II\rVert_1$ is the sum of the absolute values of the eigenvalues of the second fundamental form $II$ of $\partial K$). Moreover, for any $δ>0$, if $K$ is sufficiently $C^1$-close to a ball, we show the almost sharp Minkowski inequality $$ \left(\int_{\partial K} H^+ d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge (C_1(n)-δ)Per(K)^{\frac1{n-1}} . $$ If $K$ is axially symmetric, we prove the Minkowski inequality with the sharp constant (i.e., $δ=0$). We establish also the sharp quantitative stability (in the family of $C^1$-perturbations of the ball) of the volumetric Minkowski inequality $$ \left(\int_{\partial K} H^+ d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_2(n)|K|^{\frac1n} , $$ where $C_2(n)$ is the constant that yields the equality when $K=B_1$. Finally, we show, by constructing a counterexample, that the mentioned inequalities are false (even for domains $C^1$-close to the ball) if one replaces $H^+$ with $H$.
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
We study the asymptotic behaviour of the expected cost of the random matching problem on a $2$-dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2-dimensional matching problem). In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2-dimensional closed manifold. We also sharpen the estimate of the error term given by M. Ledoux (On optimal matching of Gaussian samples II) for the semi-discrete matching. As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.
Given $n\geq 3$, consider the critical elliptic equation $Δu + u^{2^*-1}=0$ in $\mathbb R^n$ with $u > 0$. This equation corresponds to the Euler-Lagrange equation induced by the Sobolev embedding $H^1(\mathbb R^n)\hookrightarrow L^{2^*}(\mathbb R^n)$, and it is well-known that the solutions are uniquely characterized and are given by the so-called ``Talenti bubbles''. In addition, thanks to a fundamental result by Struwe, this statement is ``stable up to bubbling'': if $u:\mathbb R^n\to(0,\infty)$ almost solves $Δu + u^{2^*-1}=0$ then $u$ is (nonquantitatively) close in the $H^1(\mathbb R^n)$-norm to a sum of weakly-interacting Talenti bubbles. More precisely, if $δ(u)$ denotes the $H^1(\mathbb R^n)$-distance of $u$ from the manifold of sums of Talenti bubbles, Struwe proved that $δ(u)\to 0$ as $\lVertΔu + u^{2^*-1}\rVert_{H^{-1}}\to 0$. In this paper we investigate the validity of a sharp quantitative version of the stability for critical points: more precisely, we ask whether under a bound on the energy $\lVert\nabla u\rVert_{L^2}$ (that controls the number of bubbles) it holds $δ(u) \lesssim \lVertΔu + u^{2^*-1}\rVert_{H^{-1}}$. A recent paper by the first author together with Ciraolo and Maggi shows that the above result is true if $u$ is close to only one bubble. Here we prove, to our surprise, that whenever there are at least two bubbles then the estimate above is true for $3\le n\le 5$ while it is false for $n\ge 6$. To our knowledge, this is the first situation where quantitative stability estimates depend so strikingly on the dimension of the space, changing completely behavior for some particular value of the dimension $n$.
We show that, on a $2$-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $L^2$-norm by identity plus the gradient of the solution to the Poisson problem $-Δf^{n,t} = μ^{n,t}-1$, where $μ^{n,t}$ is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of Caracciolo et al. (Scaling hypothesis for the Euclidean bipartite matching problem) is strong enough to capture the behavior of the optimal map in addition to the value of the optimal matching cost. As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.
Pushing a little forward an approach proposed by Villani, we are going to prove that in the Riemannian setting the condition $\nabla^2 f< g$ implies that $f$ is $c$-concave with respect to the quadratic cost as soon as it has a sufficiently small $C^1$-norm. From this, we deduce a sufficient condition for the optimality of transport maps.