A Note on Koldobsky's Lattice Slicing Inequality

$ 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*} for some absolute constant $cu003e 0$, where $vec{y}^perp$ denotes the subspace orthogonal to $vec{y}$. This gives a partial answer to a question by Koldobsky.