Publication: A discrete approach to Zhang’s projection inequality
Authors
Alonso Gutiérrez, David ; Lucas Marín, Eduardo ; Martín Goñí, Javier
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Publisher
Cambridge University Press. Canadian Mathematical Society
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DOI
https://doi.org/10.4153/S0008414X25101624
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info:eu-repo/semantics/article
Description
Abstract
In this paper we will provide a new proof of the fact that for any convex body K ⊆ Rn 2n n n nn∞rn−1voln(K ∩ (ren + K))dr ≤0(voln(K))n+1,(voln−1(P ⊥ (K)))nn where (ei)n denotes the canonical orthonormal basis in Rn, P ⊥ (K) denotes n the orthogonal projection of K onto the linear hyperplane orthogonal to en, and volk 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.
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Citation
Alonso-Gutiérrez D, Lucas Marín E, Martín Goñi J. A discrete approach to Zhang’s projection inequality. Canadian Journal of Mathematics. Published online 2025:1-47. doi:10.4153/S0008414X25101624
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