Richard P. Stanley Seminar in Combinatorics: Unbalancing unit vectors

The Komlós conjecture is a prominent vector balancing problem: how can we choose signs to keep a sum of vectors small? Here we ask how large a signed sum can be guaranteed. Given n unit vectors in d-dimensional Euclidean space, with n at least d, we show that some choice of signs produces a sum of length at least √(2n − d). This resolves a conjecture of Ambrus and Nietert when n = d + 1. Our proof draws on a combinatorial form of Vaaler’s cube slicing theorem. We also characterize all configurations for which the bound is attained. This is joint work with Jeck Lim and Skand Parvatikar.

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