A discrete approach to Zhang’s projection inequality

Alonso-Gutiérrez, David (Universidad de Zaragoza) ; Lucas Marín, Eduardo ; Martín Goñi, Javier

Resumen: In this paper we will provide a new proof of the fact that for any convex body $K\subseteq\R^n$
$$
\frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\vol_n(K\cap(re_n+K))dr\leq\frac{(\vol_n(K))^{n+1}}{(\vol_{n-1}(P_{e_n^\perp}(K)))^n},
$$
where $(e_i)_{i=1}^n$ denotes the canonical orthonormal basis in $\R^n$, $P_{e_n^\perp}(K)$ denotes the orthogonal projection of $K$ onto the linear hyperplane orthogonal to $e_n$, and $\vol_k$ denotes the $k$-dimensional Lebesgue measure. This inequality was proved by Gardner and Zhang and it implies Zhang's inequality. We will use our new approach to this inequality in order to prove discrete analogues of this inequality and of an equivalent version of it, where we will consider the lattice point enumerator measure instead of the Lebesgue measure, and show that from such discrete analogues we can recover the aforementioned inequality and, therefore, Zhang's inequality.

Idioma: Inglés

DOI: 10.4153/S0008414X25101624

Año: 2025

Publicado en: CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES (2025), [47 pp.]
ISSN: 0008-414X

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-23R
Financiación: info:eu-repo/grantAgreement/ES/MICINN/PID2022-137294NB-I00

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)