Sets of unit vectors with small subset sums

TL;DR

This paper investigates the maximum size of families of unit vectors in Banach spaces with small subset sums, providing exact values and bounds for various parameters using combinatorial and geometric tools.

Contribution

It establishes the exact value of $CB(k,d)$ for all $k,d extgreater 1$ and derives bounds for $C(k,d)$, advancing understanding of vector families with collapsing sum conditions.

Findings

$CB(k,d)= ext{max} ext{ }k+1, 2d$ for all $k,d extgreater 1$

Derived bounds and exact values for $C(k,d)$ in various cases

Applied combinatorial, convexity, and linear algebra techniques

Abstract

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a $k$-collapsing family of unit vectors in a $d$\dimensional Banach space, where the maximum is taken over all spaces of dimension $d$. Similarly, let $CB(k,d)$ denote the maximum cardinality if we require in addition that $\sum_{i=1}^m x_i=o$. The case $k=2$ was considered by F\"uredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawlor and Morgan (1994) on geometric shortest networks in smooth finite-dimensional Banach spaces. We show that $CB(k,d)=\max{k+1,2d}$ for all $k,d\geq 2$. The behaviour of $C(k,d)$ is not as simple, and we derive various upper and lower bounds for various ranges of $k$ and $d$. These include the exact values $C(k,d)=\max{k+1,2d}$ in certain cases. We use a variety of tools from graph theory, convexity and linear algebra in the proofs: in particular the Hajnal-Szemer\'edi Theorem, the Brunn-Minkowski inequality, and lower bounds for the rank of a perturbation of the identity matrix.