arXiv subjects

Joaquín Singer

Publications and source records attributed to Joaquín Singer.

6 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

Sampling properties of the zeroes of the Gaussian entire function

We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallstén's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets that are not necessarily sampling. On the probabilistic front, we estimate the number of zeroes of the Gaussian entire functions that are close to each other. We use these to prove random sampling inequalities for polynomials of degree at most $d$ using ${d}+o(d)$ points, and show that, with high probability, the sampling constants grow slower than $d^\varepsilon$ for any $\varepsilon>0$. In particular, we recover a result from Lyons and Zhai in the case of the Gaussian entire function, where it is shown that the zeroes are (almost surely) a uniqueness set for the Fock space.

math.PR↗

arXiv2024

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 $\exp\left(Ω(\sqrt{n})\right)$. Further, for $ψ_2$ bodies they proved that at most $\exp(-Ω(n))\cdot4^n$ translations of its interior are needed to cover it. Through a probabilistic approach we show that the bound $\exp(-Ω(n))\cdot4^n$ 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↗

arXiv2021

A look into homomorphisms between uniform algebras over a Hilbert space

We study the vector-valued spectrum $\mathcal{M}_{u,\infty}(B_{\ell_2},B_{\ell_2})$ which is the set of nonzero algebra homomorphisms from $\mathcal{A}_u(B_{\ell_2})$ (the algebra of uniformly continuous holomorphic functions on $B_{\ell_2}$) to $\mathcal {H}^\infty(B_{\ell_2})$ (the algebra of bounded holomorphic functions on $B_{\ell_2}$). This set is naturally projected onto the closed unit ball of $\mathcal {H}^\infty(B_{\ell_2}, \ell_2)$ giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valued cluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtaining results regarding the existence of analytic balls into these sets.

math.FA↗

arXiv2019

Homomorphisms between algebras of holomorphic functions on the infinite polydisk

We study the vector-valued spectrum $\mathcal{M}_\infty(B_{c_0},B_{c_0})$, that is, the set of non null algebra homomorphisms from $\mathcal H^\infty(B_{c_0})$ to $\mathcal H^\infty(B_{c_0})$ which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_{c_0}, \ell_\infty)$, likewise the scalar-valued spectrum $\mathcal M_\infty(B_{c_0})$ which is projected over $\bar{B}_{\ell_\infty}$. Our itinerary begins in the scalar-valued spectrum $\mathcal{M}_\infty(B_{c_0})$: by expanding a result by Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$ disjoint analytic Gleason isometric copies of $B_{\ell_\infty}$. For the vector-valued case, building on the previous result we obtain $2^c$ disjoint analytic Gleason isometric copies of $B_{\mathcal{H}^\infty(B_{c_0},\ell_\infty)}$ on each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra $\mathcal{M}_{u,\infty}(B_{c_0},B_{c_0})$ and $\mathcal{M}_\infty(B_{c_0},B_{c_0})$.

math.FA↗

arXiv2019

A fibered description of the vector-valued spectrum

For Banach spaces $X$ and $Y$ we study the vector-valued spectrum $\mathcal M_\infty(B_X,B_Y)$, that is the set of non null algebra homomorphisms from $\mathcal H^\infty(B_X)$ to $\mathcal H^\infty(B_Y)$, which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_Y, X^{**})$. The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.

math.FA↗