arXiv subjects

Damián Pinasco

Publications and source records attributed to Damián Pinasco.

17 recordsLinked to original sources

arXiv2026

Almost norming vertices for homogeneous polynomials on the cube

Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $γ_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely the cube $[-1,1]^n$, can be recovered from its values at the vertices. For $P\in\mathcal{P}_m(\mathbb{R}^n)$, set \[ M(P)=\max_{x\in[-1,1]^n}|P(x)|, \qquad V(P)=\max_{\varepsilon\in\{-1,1\}^n}|P(\varepsilon)|. \] If $P_n$ is chosen according to $γ_{m,n}$, we prove that the relative loss \[ 1-\frac{V(P_n)}{M(P_n)} \] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<β<1/2$, \[ γ_{m,n}\left\{ P\in\mathcal{P}_m(\mathbb{R}^n): V(P)\geq (1-n^{-β})M(P) \right\} \longrightarrow1 \] as $n\to\infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.

math.FA↗

arXiv2026

The weak maximizing property, compact perturbations and duality

We construct a one-parameter family $(X_c)_{0<c<1}$ of real reflexive Banach spaces, isomorphic to $\ell_2$ and satisfying $d_{\mathrm{BM}}(X_c,\ell_2)\le c^{-1}$, such that $(X_c,X_c)$ fails the weak maximizing property while $(X_c^*,X_c^*)$ has it. The failure of the WMP for $X_c$ is witnessed by a diagonal operator which does not attain its norm but admits a non-weakly null maximizing sequence. On the dual side, a separated-block estimate yields, after dualization, a reverse Pythagorean inequality and a quantitative form of the Opial property; together with property $(M)$, this gives the weak maximizing property for $X_c^*$. We also show that the compact perturbation property is self-dual for reflexive pairs. Since the weak maximizing property implies the compact perturbation property, each $(X_c,X_c)$ has the compact perturbation property. Consequently, the compact perturbation property does not imply the weak maximizing property even arbitrarily close to Hilbert space, and the weak maximizing property is not stable under duality in the reflexive setting.

math.FA↗

arXiv2026

Large signed sums of unit vectors: the first linearly dependent case

We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given $d+1$ unit vectors $u_1,\ldots,u_{d+1}$ in $\mathbb R^d$, with $d\ge2$, the problem asks for the smallest possible value of \[ \max_{\varepsilon_i=\pm1} \left\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\|. \] We prove that this value is $\sqrt{d+2}$. We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.

math.MG↗

arXiv2026

Bang vectors in the real polarization problem: The exact dimensional range

We determine the exact dimensional range in which a normalized longest signed sum, which we call a Bang vector, always satisfies the real polarization inequality: this prescription is universally valid if and only if $n\leq 14$. For every $n\geq 15$ we construct $n$ unit vectors whose all-positive sum is, up to global sign, the unique longest signed sum, while its normalized direction has polarization product smaller than $n^{-n/2}$. The counterexamples are given by a single explicit family, and for the same configurations we exhibit nonmaximal signed sums whose normalized directions do satisfy the polarization bound.

math.FA↗

arXiv2026

Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels

We establish a potential-theoretic sufficient condition for a compact subset of the real line to admit an optimal polynomial mesh. The criterion bounds the cardinality of a norming set by a linear term in the polynomial degree plus the number of connected components of a Green sublevel at height $α/n$, where $α>0$ is fixed. By combining this principle with an estimate due to Andrievskii, we prove that every uniformly perfect compact subset of $\mathbb{R}$ admits an optimal polynomial mesh. A standard product argument then produces optimal meshes on finite Cartesian products; in particular, it yields an optimal mesh on the planar Cantor dust $C\times C$, which is self-similar, totally disconnected, and has empty interior.

math.CA↗

arXiv2026

An Improved Volume Ratio Bound via Isotropic Positions

We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.

math.MG↗

arXiv2026

Strong Polarization and Entropy

We show that for any set of $n$ unit vectors $v_1,\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\ldots,p_n$ satisfying $\sum_j p_j = 1$, there exists a unit vector $u$ such that \[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. \] This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequality for products of powers of linear functionals and a strengthening of Bang's classical plank theorem for Hilbert spaces. The proof follows the approach introduced by Martínez and Ortega-Moreno in their recent solution to the strong polarization conjecture posed by Ball and Frenkel. We further note that our weighted inequality admits a Shannon-entropy interpretation: in a random sensing model, the entropy of the weights controls the minimum expected logarithmic loss.

math.FA↗

arXiv2022

On the volume ratio of projections of convex bodies

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies $K$ and $L$ is large. Namely, we prove that for every $1\leq k\leq n$ and for each convex body $K\subset \mathbb{R}^n$ there is a centrally symmetric body $L \subset \mathbb{R}^n$ such that for any two projections $P, Q: \mathbb{R}^n \to \mathbb{R}^n$ of rank $k$ one has $$ \mbox{vr}(PK, QL) \geq c \, \min\left\{\frac{ k}{ \sqrt{n}} \, \sqrt{\frac{1}{\log \log \log(\frac{n\log(n)}{k})}}, \, \frac{\sqrt{k}}{\sqrt{\log(\frac{n\log(n)}{k})}}\right\}, $$ where $c>0$ is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime $k\geq n^{2/3}$.

math.MG↗

arXiv2020

Asymptotic estimates for the largest volume ratio of a convex body

The largest volume ratio of given convex body $K \subset \mathbb{R}^n$ is defined as $$\mbox{lvr}(K):= \sup_{L \subset \mathbb{R}^n} \mbox{vr}(K,L),$$ where the $\sup$ runs over all the convex bodies $L$. We prove the following sharp lower bound $$c \sqrt{n} \leq \mbox{lvr}(K),$$ for every body $K$ (where $c>0$ is an absolute constant). This result improves the former best known lower bound, of order $\sqrt{\frac{n}{\log \log(n)}}$. We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, we show that $\mbox{lvr}(K)$ behaves as the square root of the dimension of the ambient space in the following cases: if $K$ is the unit ball of an unitary invariant norm in $\mathbb{R}^{d \times d}$ (e.g., the unit ball of the $p$-Schatten class $S_p^d$ for any $1 \leq p \leq \infty$), $K$ is the the unit ball of the full/symmetric tensor product of $\ell_p$-spaces endowed with the projective or injective norm or $K$ is unconditional.

math.MG↗

arXiv2020

Orthant probabilities and the attainment of maxima on a vertex of a simplex

We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree $k>4$, the probability that a $k$-homogeneous polynomial in $n$ variables attains a relative maximum on a vertex of the $n$-dimensional simplex tends to one as the dimension $n$ grows. The bounds we obtain for the orthant probabilities are tight up to $\log(n)$ factors.

math.PR↗

arXiv2019

The minimal volume of simplices containing a convex body

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where $c>0$ is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if $K$ is in isotropic position, we present a method to find centered simplices verifying the above bound that works with very high probability. As a consequence, we provide correct asymptotic estimates on an old problem in convex geometry. Namely, we show that the simplex $S_{min}(K)$ of minimal volume enclosing a given convex body $K \subset \mathbb R^n$, fulfills the following inequality $$\left(\frac{vol(S_{min}(K))}{vol(K)}\right)^{1/n} \leq d \sqrt{n},$$ for some absolute constant $d>0$. Up to the constant, the estimate cannot be lessened.

math.MG↗

arXiv2017

On the linear polarization constants of finite dimensional spaces

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces $\ell_p^d$: they behave as $\sqrt[p]{d}$ if $1\le p\le 2$ and as $\sqrt{d}$ if $2\le p<\infty$. For $p=\infty$ we get the asymptotic behavior up to a logarithmic factor

math.FA↗

arXiv2015

Hypercyclic behavior of some non-convolution operators on $H(\mathbb{C}^N)$

We study hypercyclicity properties of a family of non-convolution operators defined on spaces of holomorphic functions on $\mathbb{C}^N$. These operators are a composition of a differentiation operator and an affine composition operator, and are analogues of operators studied by Aron and Markose on $H(\mathbb{C})$. The hypercyclic behavior is more involved than in the one dimensional case, and depends on several parameters involved.

math.FA↗

arXiv2015

An integral formula for multiple summing norms of operators

We prove that the multiple summing norm of multilinear operators defined on some $n$-dimensional real or complex vector spaces with the $p$-norm may be written as an integral with respect to stables measures. As an application we show inclusion and coincidence results for multiple summing mappings. We also present some contraction properties and compute or estimate the limit orders of this class of operators.

math.FA↗

arXiv2014

Energy integrals and metric embedding theory

For some centrally symmetric convex bodies $K\subset \mathbb R^n$, we study the energy integral $$ \sup \int_{K} \int_{K} \|x - y\|_r^{p}\, dμ(x) dμ(y), $$ where the supremum runs over all finite signed Borel measures $μ$ on $K$ of total mass one. In the case where $K = B_q^n$, the unit ball of $\ell_q^n$ (for $1 < q \leq 2$) or an ellipsoid, we obtain the exact value or the correct asymptotical behavior of the supremum of these integrals. We apply these results to a classical embedding problem in metric geometry. We consider in $\mathbb R^n$ the Euclidean distance $d_2$. For $0 < α< 1$, we estimate the minimum $R$ for which the snowflaked metric space $(K, d_2^α)$ may be isometrically embedded on the surface of a Hilbert sphere of radius $R$.

math.MG↗

arXiv2014

Strongly mixing convolution operators on Fréchet spaces of holomorphic functions

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on $\mathbb{C}^n$ are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy-Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.

math.FA↗

arXiv2013

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper we obtain bounds for the decay rate in the $L^r (\rr^d)$-norm for the solutions to a nonlocal and nolinear evolution equation, namely, $$u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy, $$ with $ x \in \rr^d$, $ t>0$. Here we consider a kernel $K(x,y)$ of the form $K(x,y)=ψ(y-a(x))+ψ(x-a(y))$, where $ψ$ is a bounded, nonnegative function supported in the unit ball and $a$ is a linear function $a(x)= Ax$. To obtain the decay rates we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy$, with $1 \leq p < \infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole $\rr^d$: $$ λ_{1,p} (\rr^d) = 2(\int_{\rr^d} ψ(z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. $$ Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p \to \infty$ of $λ_{1,p}^{1/p}$.

math.AP↗