Minimizing the sum of projections of a finite set

TL;DR

This paper investigates the minimal sum of projections of finite sets in Euclidean space, providing a new proof, stability results, and solutions for specific projection sums, with implications for graph isoperimetric inequalities.

Contribution

It introduces a new, constructive proof for the minimal projection sum problem, enabling stability analysis and algebraic characterization, and solves the problem for one-dimensional projections.

Findings

Established a linear order minimizing projection sums.

Derived stability results for near-optimal sets.

Solved the minimization problem for one-dimensional projections.

Abstract

Consider the projections of a finite set $A\subset R^n$ onto the coordinate hyperplanes. How small can the sum of the sizes of these projections be, given the size of $A$? In a different form, this problem has been studied earlier in the context of edge-isoperimetric inequalities on graphs, and it is can be derived from the known results that there is a linear order on the set of $n$-tuples with non-negative integer coordinates, such that the sum in question is minimised for the initial segments with respect to this order. We present a new, self-contained and constructive proof, enabling us to obtain a stability result and establish algebraic properties of the smallest possible projection sum. We also solve the problem of minimising the sum of the sizes of the one-dimensional projections.