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Rainie Bozzai: The Vector Balancing Constant for Zonotopes |
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| The vector balancing constant 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 d-dimensional zonotope K, one has vb(K , K ) =O(sqrt(d)) . Intuitively, this asks whether a natural geometric generalization of Spencer’s Theorem (for which K is the cube ) holds. We prove that for any d-dimensional zonotope K one has vb(K , K ) =O(sqrt(d) log log log 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)=O(sqrt(dlogd)). All of the bounds are constructive and the corresponding colorings can be computed in polynomial time. The lecture will be in the Riesz lecture room. |
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