A Nearly Quadratic Lower Bound for Linear Optimization over Convex Bodies in the Membership Oracle Model

The paper establishes nearly quadratic lower bounds for randomized algorithms that perform linear optimization and uniform sampling over convex bodies using a membership oracle. It shows that the lower bound for linear optimization matches the best known upper bound up to a polylogarithmic factor in the dimension, while the bound for uniform sampling improves upon the previous linear lower bound. Additionally, the authors demonstrate that their construction yields the same lower bound for volume estimation.

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