Does exist positive constant $c_d$, depending only from dimension with the property: for every convex body $K\subset \mathbb R^d$ exists parallelepiped $P\subset K$ so that $$ |P\cap \mathbb Z^d| \ge c_d |K\cap \mathbb Z^d|? $$
The maximal discrete parallelepiped in a discrete convex body
Petr Petrukov
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