Approximating the centroid is hard
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Limits and Applications of Group Algebras for Parameterized Problems
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Volume computation using a direct monte carlo method
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
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How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in \mathbb{R}^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.