Finding a subset of nonnegative vectors with a coordinatewise large sum

TL;DR

This paper establishes a precise bound on selecting a subset of nonnegative vectors so that their sum exceeds a scaled average, using linear programming techniques, with the bound proven to be sharp for large N.

Contribution

It introduces a new bound for subset selection of nonnegative vectors with a sum exceeding a scaled average, employing Carathéodory's theorem, and proves the bound's sharpness.

Findings

Derived a formula for the subset size based on rational parameters

Proved the bound is sharp for large N

Applied linear programming techniques in the proof

Abstract

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least $(p/q)(u_1+...+u_N)$. For fixed $d$ and $a$, this bound is sharp if $N$ is large enough. The method of the proof uses Carath\'eodory's theorem from linear programming.