arXiv subjects

Julián Haddad

Publications and source records attributed to Julián Haddad.

14 recordsLinked to original sources

arXiv2026

Extensions of the Busemann-Petty Problem for Arbitrary Measures

The classical Busemann--Petty problem asks whether smaller central hyperplane sections of origin-symmetric convex bodies imply smaller total volume. Zvavitch studied the analogous question when sections and bodies are measured by two arbitrary densities. We refine this result in three directions: we allow central sections of arbitrary codimension; we relax the monotonicity requirement on the radial density ratio to a decomposition into a non-decreasing and a non-increasing part; and we permit a distinct pair of densities for each body, one for the sections and another for the full volume. We also obtain an isomorphic version, in which the comparison constant is governed by the distance from an auxiliary star body to the class of generalized $k$-intersection bodies, and we present some examples illustrating cases not covered by previous results.

math.MG↗

arXiv2025

On the polar of Schneider's difference body

In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santaló inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically á la Bourgain-Milman. We also consider a functional version.

math.MG↗

arXiv2025

Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies

In 1970, Schneider introduced the $m$th order difference body of a convex body, and also established the $m$th-order Rogers-Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang's projection inequality, Petty's projection inequality, the Busemann-Petty centroid inequality and Busemann's random simplex inequality). We also establish a new proof of Schneider's $m$th-order Rogers-Shephard inequality. As an application, a $m$th-order affine Sobolev inequality for functions of bounded variation is provided.

math.FA↗

arXiv2025

Affine Hardy--Littlewood--Sobolev inequalities

Sharp affine Hardy--Littlewood--Sobolev inequalities for functions on $\mathbb R^n$ are established, which are significantly stronger than (and directly imply) the sharp Hardy--Littlewood--Sobolev inequalities by Lieb and by Beckner, Dou, and Zhu. In addition, sharp reverse inequalities for the new inequalities and the affine fractional $L^2$ Sobolev inequalities are obtained for log-concave functions on $\mathbb R^n$.

math.MG↗

arXiv2025

Affine Fractional Sobolev and Isoperimetric Inequalities

Sharp affine fractional Sobolev inequalities for functions on $\mathbb R^n$ are established. For each $0<s<1$, the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren and Lieb. In the limit as $s\to 1^-$, the new inequalities imply the sharp affine Sobolev inequality of Gaoyong Zhang. As a consequence, fractional Petty projection inequalities are obtained that are stronger than the fractional Euclidean isoperimetric inequalities and a natural conjecture for radial mean bodies is proved.

math.MG↗

arXiv2024

A Rogers--Brascamp--Lieb--Luttinger inequality in the space of matrices

We consider convex bodies in $M_{n,m}(\mathbb R)$, the space of matrices of $n$-rows and $m$-columns. A special case of fiber-symmetrization in $M_{n,m}(\mathbb R)$ was recently introduced in [5,6]. We prove a Rogers--Brascamp--Lieb--Luttinger type inequality with respect to this symmetrization, for quasi-concave functions and provide some applications.

math.FA↗

arXiv2024

Higher-Order Lp Isoperimetric and Sobolev Inequalities

Schneider introduced an inter-dimensional difference body operator on convex bodies and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in $\mathbb R^n$ from those in $\mathbb R^n$, were replaced by inter-dimensional simplicial operators, which generate convex bodies in $\mathbb R^{nm}$ from those in $\mathbb R^{n}$ (or vice versa). In this work, we treat the $L^p$ extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary $m$-dimensional convex bodies containing the origin. We establish $m$th-order $L^p$ isoperimetric inequalities, including the $m$th-order versions of the $L^p$ Petty projection inequality, $L^p$ Busemann-Petty centroid inequality, $L^p$ Santaló inequalities, and $L^p$ affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals $(\mathbb R^n, \|\cdot\|_E) \to (\mathbb R^m, \|\cdot\|_F)$.

math.MG↗

arXiv2023

Radon transforms with small derivatives and distance inequalities for convex bodies

Generalizing the slicing inequality for functions on convex bodies from [11], it was proved in [4] that there exists an absolute constant $c$ so that for any $n\in \mathbb N$, any $q\in [0,n-1)$ which is not an odd integer, any origin-symmetric convex body $K$ of volume one in $\mathbb R^n$ and any infinitely smooth probability density $f$ on $K$ we have $$\max_{ξ\in S^{n-1}} {\frac 1{\cos(πq/2)}\mathcal R f(ξ, \cdot)_t^{(q)}(0)} \ge \left( \frac {c(q+1)}{n}\right)^{\frac{q+1}2}.$$ Here $\mathcal R f(ξ,t)$ is the Radon transform of $f$, and the fractional derivative of the order $q$ is taken with respect to the variable $t\in \mathbb R$ with fixed $ξ\in S^{n-1}.$ In this note we show that there exist an origin-symmetric convex body $K$ of volume 1 in $\mathbb R^n$ and a continuous probability density $g$ on $K$ so that $$\max_{ξ\in S^{n-1}} {\frac 1{\cos(πq/2)}\mathcal R g(ξ, \cdot)_t^{(q)}(0)} \leq \frac 1{\sqrt n} (c(q+1))^{\frac{q+1}2}.$$ In the case $q=0$ this was proved in [5,6], and it was used there to obtain a lower estimate for the maximal outer volume ratio distance from an arbitrary origin-symmetric convex body $K$ to the class of intersection bodies. We extend the latter result to the class $L_{-1-q}^n$ of bodies in $\mathbb R^n$ that embed in $L_{-1-q}.$ Namely, for every $q\in [0,n)$ there exists an origin-symmetric convex body $K$ in $\mathbb R^n$ so that ${d_{\operatorname{ovr}}}(K, L_{-1-q}^n) \ge c n^{\frac 1{2(q+1)}}.$

math.FA↗

arXiv2022

Affine fractional $L^p$ Sobolev inequalities

Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of the affine $L^p$ Sobolev inequalities of Lutwak, Yang, and Zhang. In addition, affine fractional asymmetric $L^p$ Sobolev inequalities are established.

math.MG↗

arXiv2020

From affine Poincaré inequalities to affine spectral inequalities

Given a bounded open subset $Ω$ of $\mathbb R^n$, we establish the weak closure of the affine ball $B^{\mathcal A}_p(Ω) = \{f \in W^{1,p}_0(Ω):\ \mathcal E_p f \leq 1\}$ with respect to the affine functional $\mathcal E_pf$ introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in $L^p(Ω)$ for any $p \geq 1$. These points use strongly the celebrated Blaschke-Santaló inequality. As counterpart, we develop the basic theory of $p$-Rayleigh quotients in bounded domains, in the affine case, for $p\geq 1$. More specifically, we establish $p$-affine versions of the Poincaré inequality and some of their consequences. We introduce the affine invariant $p$-Laplace operator $Δ_p^{\mathcal A} f$ defining the Euler-Lagrange equation of the minimization problem of the $p$-affine Rayleigh quotient. We also study its first eigenvalue $λ^{\mathcal A}_{1,p}(Ω)$ which satisfies the corresponding affine Faber-Krahn inequality, this is that $λ^{\mathcal A}_{1,p}(Ω)$ is minimized (among sets of equal volume) only when $Ω$ is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator $Δ_p^{\mathcal A} f$. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for $p \geq 1$. All affine inequalities obtained are stronger and directly imply the classical ones.

math.AP↗

arXiv2016

A Morse deformation lemma at infinity

We prove a generalized version of the classic deformation lemma from Morse Theory that considers functions going to $-\infty$ at a compact set, and allowing the lower value of the deformation to be $-\infty$. The result is valid for a class of functions satisfying a suitable growth condition.

math.DG↗

arXiv2016

The sharp affine $L^2$ Sobolev trace inequality and variants

We establish a sharp affine $L^p$ Sobolev trace inequality by using the $L_p$ Busemann-Petty centroid inequality. For $p = 2$, our affine version is stronger than the famous sharp $L^2$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all cases of equality in this case. For this new inequality, no Euclidean geometric structure is needed.

math.FA↗

arXiv2013

Continuous dependence on the derivative of generalized heat equations

We consider here a generalized heat equation $\partial_t ρ=\frac{d}{dx}\frac{d}{dW}ρ$, where $W$ is a finite measure on the one dimensional torus, and $\frac{d}{dW}$ is the Radon-Nikodym derivative with respect to $W$. Such equation has appeared in different contexts, being related to physical systems and representing a large class of classical and non-classical parabolic equations. As a natural assumption on $W$, we require that the Lebesgue measure is absolutely continuous with respect to $W$. The main result here presented consists in proving, for a suitable topology, a continuous dependence of the solution $ρ$ as a function of $W$.

math.AP↗

arXiv2013

On existence of periodic solutions for Kepler type problems

We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In the different cases, the lower bounds obtained for the number of solutions are related to the winding number of a curve in the plane, the homology of a space in $\R^3$, the knot type of a curve and the link type of a set of curves. Also, the results are applied to the restricted $n$-body problem.

math.CA↗