arXiv subjects

Daniel Galicer

Publications and source records attributed to Daniel Galicer.

At least 19 recordsLinked to original sources

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

Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes

We investigate local invariants and geometric phenomena for polynomial spaces of low degree on the $q$-ary Hamming scheme $C_q^N$, where $C_q$ denotes the cyclic group of order $q$. Our main analytic tool is a support-sensitive Bohnenblust--Hille inequality for spherical polynomial spaces, showing that the relevant complexity parameter is the support size of the monomials rather than their total degree. Equivalently, in the corresponding toroidal formulation, this leads to estimates for polynomials whose coordinate degrees are bounded by $q-1$, while the growth of the constants is governed by the interaction order of the variables. These inequalities yield applications to the learning theory of spherical low-level functions and also provide the basis for dimension-free comparisons between several classical local invariants, including Sidon constants, unconditional basis constants, and Gordon--Lewis constants. As a consequence, we obtain sharp asymptotic estimates for these invariants in the spherical setting, with analogous comparison and asymptotic results for homogeneous and tetrahedral polynomial spaces. We also study projection constants and the associated reproducing kernels. In the spherical case, suitably normalized Krawtchouk polynomials converge to Hermite polynomials under central-limit scaling, leading to explicit Gaussian limits and sharp asymptotic formulas. By contrast, in the homogeneous and tetrahedral settings a dichotomy appears between the Boolean case and the regime $q\ge3$, where the limiting behaviour is governed by moments of a circular complex Gaussian.

math.FA↗

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 Mart\'inez 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↗

arXiv2025

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

Local constants and Bohr's phenomenon for Banach spaces of analytic polynomials

The primary aim of this work is to develop methods that provide new insights into the relationships between fundamental constants in Banach space theory--specifically, the projection constant, the unconditional basis constant and the Gordon-Lewis constant--for the Banach space $\mathcal{P}_J(X_n)$ of multivariate analytic polynomials. This class consists of all polynomials whose monomial coefficients vanish outside the set of multi-indices $J$, and it is equipped with the supremum norm on the unit sphere of the finite-dimensional Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$. We establish a~general framework for proving quantitative results on the asymptotic optimal behavior of these constants, which depend on both the dimension of the space and the degree of the polynomials. Using the tools developed, we derive asymptotic estimates of the Bohr radius for general Banach sequence lattices. Additionally, we apply our results to the asymptotic study of local constants and the Bohr radius within finite-dimensional Lorentz sequence spaces, which requires a~refined analysis of the combinatorial structure of the associated index sets. As a consequence, we obtain optimal results across a broad range of parameters.

math.FA↗

arXiv2024

Ryll-Wojtaszczyk Formulas for bihomogeneous polynomials on the sphere

We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm projection aligns with the natural orthogonal projection. This result enables us to establish a connection between these constants and weighted \linebreak $L_1$-norms of specific Jacobi polynomials. Consequently, we derive explicit bounds, provide practical expressions for computation, and present asymptotically sharp estimates for these constants. Our findings extend the classical Ryll and Wojtaszczyk formula for the projection constant of homogeneous polynomials in finite-dimensional complex Hilbert spaces to the bihomogeneous setting.

math.FA↗

arXiv2024

The $p$-Operator Approximation Property

We study a notion analogous to the $p$-Approximation Property ($p$-AP) for Banach spaces, within the noncommutative context of operator spaces. Referred to as the $p$-Operator Approximation Property ($p$-OAP), this concept is linked to the ideal of operator $p$-compact mappings. We present several equivalent characterizations based on the density of finite-rank mappings within specific spaces for different topologies, and also one in terms of a slice mapping property. Additionally, we investigate how this property transfers from the dual or bidual to the original space. As an application, the $p$-OAP for the reduced $C^*$-algebra of a discrete group implies that operator $p$-compact Herz-Schur multipliers can be approximated in $\mbox{cb}$-norm by finitely supported multipliers.

math.FA↗

arXiv2024

Revisiting Operator $p$-Compact Mappings

We continue our study of the mapping ideal of operator $p$-compact maps, previously introduced by the authors. Our approach embraces a more geometric perspective, delving into the interplay between operator $p$-compact mappings and matrix sets, specifically we provide a quantitative notion of operator $p$-compactness for the latter. In particular, we consider operator $p$-compactness in the bidual and its relation with this property in the original space. Also, we deepen our understanding of the connections between these mapping ideals and other significant ones (e.g., completely $p$-summing, completely $p$-nuclear).

math.FA↗

arXiv2024

Minimal projections onto spaces of polynomials on real euclidean spheres

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted $L_1$-norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions and asymptotically accurate estimates for them. The real case we address is considerably more nuanced than its complex counterpart.

math.FA↗

arXiv2023

Hadwiger's problem for bodies with enough sub-Gaussian marginals

Hadwiger's conjecture in convex geometry, formulated in 1957, states that every convex body in $\mathbb{R}^n$ can be covered by $2^n$ translations of its interior. Despite significant efforts, the best known bound related to this problem was $\mathcal{O}(4^n \sqrt{n} \log n)$ for more than sixty years. In 2021, Huang, Slomka, Tkocz, and Vritsiou made a major breakthrough by improving the estimate by a factor of $\exp\left(\Omega(\sqrt{n})\right)$. Further, for $\psi_2$ bodies they proved that at most $\exp(-\Omega(n))\cdot4^n$ translations of its interior are needed to cover it. Through a probabilistic approach we show that the bound $\exp(-\Omega(n))\cdot4^n$ can be obtained for convex bodies with sufficiently many well-behaved sub-gaussian marginals. Using a small diameter approximation, we present how the currently best known bound for the general case, due to Campos, Van Hintum, Morris, and Tiba can also be deduced from our results.

math.MG↗

arXiv2023

Operator space tensor norms

The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. The aim of this manuscript is to kick off the development of a systematic theory of tensor products and tensor norms for operator spaces and its interplay with their associated mapping ideals. Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definitions in the operator space framework. The theory is not a mere translation of what is known in the classical setting and new insights, techniques, ideas or hypotheses are required in many cases. As a consequence, notable differences in the theory appear when compared to the classical one.

math.OA↗

arXiv2023

Projection constants for spaces of Dirichlet polynomials

Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(\omega)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $\omega$-Dirichlet groups, we prove the formula $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-i\omega_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $\omega$ and index sets $J$. To see an example, combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space $\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $

math.FA↗

arXiv2023

Asymptotic insights for projection, Gordon-Lewis and Sidon constants in Boolean cube function spaces

The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^N$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^N$ that have Fourier--Walsh expansions supported on a fixed~family $\mathcal{S}$ of subsets of $\{1, \ldots, N\}$. Our investigation centers on the projection, Sidon and Gordon--Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants.

math.FA↗

arXiv2023

The projection constant for the trace class

We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $ \boldsymbol{\lambda}\big(\mathcal S_1(n)\big) = n \int_{\mathcal U_n} \vert \text{tr}(U) \vert \,dU \,, $ where the integration is with respect to the Haar probability measure on the group $\mathcal U_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula $ \lim_{n\to \infty} \boldsymbol{\lambda}\big(\mathcal S_1(n)\big)/n = \sqrt{\pi}/2\,. $

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↗

arXiv2022

Projection constants for spaces of multivariate polynomials

The general problem we address is to develop new methods in the study of projection constants of Banach spaces of multivariate polynomials. The relative projection constant $\boldsymbolλ(X,Y)$ of a subspace $X$ of a Banach $Y$ is the smallest norm among all possible projections on $Y$ onto $X$, and the projection constant $\boldsymbolλ(X)$ is the supremum of all relative projection constants of $X$ taken with respect to all possible super spaces $Y$. This is one of the most significant notions of modern Banach space theory and has been intensively studied since the birth of abstract operator theory. We focus on projection constants of Banach spaces of multivariate polynomials formed either by trigonometric polynomials $f(g)=\sum_{γ\in E} \hat{f}(γ) γ(g)$ defined on a compact topological group $G$, which have Fourier coefficients $\hat{f}(γ)$ supported in a finite set $E$ of characters; or analytic polynomials $P(z)=\sum_{α\in J}c_α(P)\,z^α$, which are defined on a Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$ and have monomial coefficients $c_α(P)$ supported in a finite set $J \subset \mathbb{N}_0^n$ of multi indices. Depending on the underlying structure (of the group, Banach space or index set), the goal is to prove precise formulas or asymptotically optimal estimates. Our general setting is flexible enough to handle a wide variety of Banach spaces of polynomials, including analytic polynomials on polydiscs, Dirichlet polynomials on the complex plane, and polynomials on Boolean cubes $\{-1,+1\}^n$. Moreover, we get an explicit formula for the projection constant of the space of trace class operators. The methods developed here enable us to prove new estimates for important invariants such as the unconditional basis constant and the Gordon-Lewis constant for Banach spaces of multivariate polynomials.

math.FA↗

arXiv2022

Multipliers for Hardy spaces of Dirichlet series

We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its associated multiplication operator. In particular, we study the norm, the essential norm, and the spectrum for an operator of this kind. We exploit the existing natural identification of spaces of Dirichlet series with spaces of holomorphic functions in infinitely many variables and apply several methods from complex and harmonic analysis to obtain our results. As a byproduct we get analogous statements on such Hardy spaces of holomorphic functions.

math.CV↗

arXiv2021

Continuous quantitative Helly-type results

Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland and Youssef. This, together with an appropriate selection in the accuracy of the approximation, allow us to obtain Helly-type versions which are sensitive to the number of convex sets involved.

math.MG↗