Minimizing the sum of projections of a finite set

Vsevolod F. Lev, Misha Rudnev

Published 2016-10-08Version 1

Consider the projections of a finite set $A\subset\mathbb R^n$ onto the coordinate hyperplanes; how small can the sum of the sizes of these projections be, given the size of $A$? We introduce a linear order on the set of $n$-tuples with non-negative integer coordinates, such that the sum in question is minimized for the initial segments with respect to this order. Our results support the intuition that the sum is minimized when $A$ is close to an $n$-dimensional cube.