Since the development of the first randomized polynomial-time algorithm for volume computation by Dyer, Frieze, and Kannan in 1989, convex-body sampling has been a central problem at the intersection of algorithms, geometry, and probability. A major milestone came in 1997, when Kannan, Lovász, and Simonovits analyzed the Ball Walk and formulated the influential KLS conjecture. This was extended to log-concave distributions by Lovász and Vempala in 2006, and further accelerated by Cousins and Vempala in 2015 through warm-start generation techniques.
In this talk, I will present a gentle introduction to these milestones and how they have been streamlined and improved in the past few years. The talk is based on joint work with Santosh Vempala.
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