Sets of unit vectors with small subset sums

We say that a family of m {xi}Ιi ε[m]\} vectors in a Banach space X satisfies the k-collapsing condition if the sum of any k of them has norm at most 1. 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 the m vectors sum to 0. The case k = 2 was considered by Füredi, 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 ≥ 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–Szemerédi Theorem, the Brunn– Minkowski inequality, and lower bounds for the rank of a perturbation of the identity matrix.