The Vector Balancing Constant for Zonotopes
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The vector balancing constant vb(K,Q) of two symmetric convex bodies K,Q is the minimum r ≥ 0 so that any number of vectors from K can be balanced into an r-scaling of Q. A question raised by Schechtman is whether for any zonotope K ⊆ℝ^d one has vb(K,K) ≲√(d). Intuitively, this asks whether a natural geometric generalization of Spencer's Theorem (for which K = B^d_∞) holds. We prove that for any zonotope K ⊆ℝ^d one has vb(K,K) ≲√(d)logloglog d. Our main technical contribution is a tight lower bound on the Gaussian measure of any section of a normalized zonotope, generalizing Vaaler's Theorem for cubes. We also prove that for two different normalized zonotopes K and Q one has vb(K,Q) ≲√(d log d). All the bounds are constructive and the corresponding colorings can be computed in polynomial time.
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