Let \(a_1, \dots , a_n\) be vectors in \(\mathbb {R}^d\), each of length at least 1, and let \(R_1, \dots , R_k\) be \(k\) open regions of \(\mathbb {R}^d\), where \(|x - y| {\lt} 2\) for any \(x, y\) that lie in the same region \(R_i\). Then the number of linear combinations \(\sum _{i=1}^{n} \epsilon _i a_i\), \(\epsilon _i \in \{ 1, -1\} \), that can lie in the union \(\bigcup _i R_i\) of the regions is at most the sum of the \(k\) largest binomial coefficients \(\binom {n}{j}\).

In particular, we get the bound \(\binom {\lfloor n/2 \rfloor }{n}\) for \(k = 1\).