70 problems

Let $\mu$ be an isotropic log-concave probability measure on $\mathbb{R}^n$, and let $C_P(\mu)$ denote its Poincaré constant, namely the least constant for which the Poincaré inequ…

Bourgain's hyperplane conjecture. There exists a universal constant $c>0$, independent of $n$, such that

Log-Sobolev KLS conjecture.

KLS conjecture. There exists a universal constant $\mathbf{C}_{\mathrm{univ}}>0$ such that

Thin-shell conjecture. The sequence $(S_n)$ is bounded.

For each $n$, let $\mu_n$ be a log-concave probability measure on $\mathbb{R}^n$, let $\Lambda_{\mu_n}^\ast$ denote its Cramér transform, and let $\mathcal{T}_n$ be … A family exhi…

Concavity conjecture. The function $f$ is concave on $[0,1]$. This conjecture proposes a stronger form of regularity for entropy under normalized sums of independent identically di…

Latała's moment conjecture. There exists a universal constant $C$ such that, for every log-concave vector $X$ with values in a finite-dimensional normed space $(F,\|\ \|)$ and ever…

Let $X$ be an isotropic unconditional log-concave random vector in $\mathbb{R}^n$, meaning that its distribution is invariant under coordinate reflections and it has a logarithmica…

Let $K\subset\mathbb{R}^n$ be an isotropic convex body, let $L_K$ denote its isotropic constant, and let $X$ be uniformly distributed on $K$. An orthonormal basis conjecture for is…

Let $X\sim\mu$, where $\mu$ is a log-concave measure on $\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial of degree $d$. For a Borel measure $\nu$ on…

Let $d\in\mathbb{N}$. Let $H_n$ be the hyperoctahedral group acting on $\mathbb{R}^n$ by coordinate permutations and sign changes. A measure $\mu$ on $\mathbb{R}^n$ is isotropic if…

A measure $\mu$ on $\mathbb{R}^n$ is log-concave if, for every convex compact sets $K,L\subset\mathbb{R}^n$ and every $t\in[0,1]$, it satisfies … The measure is even when…

Let $\mu$ be an even log-concave probability measure on $\mathbb{R}^n$, and let $K,L$ be origin-symmetric convex bodies. The concavity exponent $p(\mu)$ is defined as the largest v…

Let $Z$ be a $d$-dimensional log-concave random vector. The Sudakov minoration principle gives a lower bound for the expected supremum of a process indexed by a finite set of point…

Even log-concave dimensional inequality conjecture. An improvement of

KLS conjecture. The quantity $\Psi_n$ is bounded above by a universal constant, independent of the dimension $n$.

Let $\Omega\subseteq\mathbb R^k$ be a convex set, let $V$ be a convex function on $\Omega$, and define the weighted Neumann operator … Let the first nonconstant eigenfunction of…

Generalized B-conjecture. The function

Let $\nu=e^{-W}$ be a probability measure on $\mathbb{R}^d$, where $W$ is convex and $\|\operatorname{cov}(\nu)\|_{op}\leq 1$. An exponential concentration inequality with paramete…

For each $n\in\mathbb{N}$, let $X_n$ be the rotationally invariant random vector denoted by $\mathcal{E}$ in $\mathbb{R}^n$, and let $\Lambda_{X_n}^\ast$ be its Cramér transform. N…

Rotationally invariant sharp separability conjecture. The upper bound from the theorem for independent symmetric log-concave components should remain valid for $X$:

Let $\nu$ be an even log-concave measure on $\mathbb{R}^{n}$. Let $K_{0},K_{1}\subseteq\mathbb{R}^{n}$ be convex and origin-symmetric, meaning $K_{0}=-K_{0}$ and $K_{1}=-K_{1}$. Ev…

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$. It is $\beta$-tilt-stable if, for every $h\in\mathbb R^n$, the covariance operator of the tilted measure…

Let $\mu$ be an even log-concave measure on $\mathbb{R}^n$, let $\lambda\in(0,1)$, and let $K$ and $C$ be origin-symmetric convex bodies. Colesanti–Livshyts–Marsiglietti conjecture…