A Note on Koldobsky's Lattice Slicing Inequality

arXiv: Metric Geometry, 2016.

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Abstract:

$ newcommand{R}{{mathbb{R}}} newcommand{Z}{{mathbb{Z}}} renewcommand{vec}[1]{{mathbf{#1}}} $We show that if $K subset R^d$ is an origin-symmetric convex body, then there exists a vector $vec{y} in Z^d$ such that begin{align*} |K cap Z^d cap vec{y}^perp| / |K cap Z^d| ge min(1,c cdot d^{-1} cdot mathrm{vol}(K)^{-1/(d-1)}) ; , end{align*} f...More

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