Year: 2025
Type: article
Abstract: In this article, we will provide a new proof of the fact that for any convex body $K\subseteq \mathbb {R}^n$ , $$ \begin{align*}\frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\mathrm{vol}_n(K\cap(re_n+K))dr\leq\frac{(\mathrm{vol}_n(K))^{n+1}}{(\mathrm{vol}_{n-1}(P_{e_n^\perp}(K)))^n}, \end{align*} $$ where $(e_i)_{i=1}^n$ denotes the canonical orthonormal basis in $\mathbb {R}^n$ , $P_{e_n^\perp }(K)$ denotes the orthogonal projection of K onto the linear hyperplane orthogonal to $e_n$ , and $\mathrm {vol}_k$ 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 analogs 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 analogs we can recover the aforementioned inequality and, therefore, Zhang’s inequality.
Source: Canadian Journal of Mathematics
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
Awards: G1175115578, G121809054, G1566263632, G1855990336, G2262748287, G2309174360, G2660406951, G279363373, G300979063, G34346518, G3480869486, G3609907439, G425668148, G430777504, G5080475149, Topics in Asymptotic Geometric Analysis and Applications, G5421974903, G7084143925, G7264301175, G8920660035, G897484643, G99296377
OpenAlex ID: W4414480902
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