Together with Bourgain's slicing problem and the variance conjecture, the Kannan-Lov\'asz-Simonovits (KLS) conjecture has been a driving force for mathematical developments in high dimensional geometry in recent decades. We present here a proof of this conjecture. The crucial step in the proof is the very recent criter...
Pierre Bizeul, Bo'az Klartag, Joseph Lehec· 5 citations· ⚡2
We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given...
We present a proof that determines the optimal value of the universal constant in the thin-shell theorem for log-concave distributions in high dimensions. We prove that for any log-concave random vector $X = (X_1,\ldots,X_n)$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ {\rm Var}( |X|^2 ) \leq 8 n. $$ T...
Yuansi Chen, Bo'az Klartag· 9 citations· ⚡3
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