34 problems

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…

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

Let $\nu$ be an even log-concave measure on $\mathbb{R}^{n}$. For origin-symmetric convex bodies $K_{0},K_{1}\subseteq\mathbb{R}^{n}$ and $t\in[0,1]$, define … The measure $\nu$ is…

Log-Sobolev KLS conjecture.

Kannan–Lovász–Simonovits conjecture. There is a universal constant $C>0$ such that

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…

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…

Madiman–Kontoyannis conjecture. The entropy satisfies

Let $p\in[0,1]$, let $\mu$ be an even log-concave measure on $\mathbb R^n$, let $K,L$ be nonempty symmetric convex sets, and let $\lambda\in[0,1]$. For $p>0$, define … with $p=0$ i…

Let $0<\alpha<1$, let $\mu=w(x)dx$ be a strictly log-concave probability measure supported on all of $\mathbb{R}^n$, with centrally symmetric density $w(-x)=w(x)$, and let $E$ be a…

Let $\mu$ be a finite, strictly log-concave measure on $\mathbb{R}^n$, and let $E$ be an isoperimetric set with volume fraction $0<\alpha<1/2$. Convex sandwich conjecture. There ar…

Let $\mu=e^{-V}dx$ be a strictly log-concave probability measure on $\mathbb{R}^n$, and let $E$ be an open isoperimetric subset with $\mu(E)=\alpha$. Lipschitz-interface conjecture…

Generalized Sudakov minoration conjecture. There is a universal constant $C>0$ such that

Fix $\kappa\in[-\infty,\frac1d]$. Let $(U,V)$ be the coordinates of a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$. Symmetric $\kappa$-concave Busemann conjecture. Fo…

Let $(U,V)$ be the coordinates of a symmetric log-concave random vector in $\mathbb{R}^2$. Ball–Nayar–Tkocz's conjecture. … This inequality is equivalent to the triangle inequality…

Ball–Nayar–Tkocz's conjecture. The function $M_1^X$ defines a norm on $\mathbb{R}^d$. The conjecture is the Shannon-entropy case of a Busemann-type theorem. The source records a la…

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…

The norm conjecture. The function $N_X$ defines a norm on $\mathbb{R}^n$. This would give a geometric formulation of entropy inequalities for one-dimensional projections of symmetr…

Let $X$ and $X'$ be independent random vectors in ${\mathbb R}^n$ drawn from the same log-concave distribution. Let $h$ denote differential entropy and define the entropy power by…

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…

B-conjecture. The function

Weak and strong moments conjecture. There exist $C,c>0$ such that for any log-concave random vector $X$ and any norm $\|\cdot\|$,

Weak thin shell conjecture. There exists $c>0$ such that for every log-concave random vector $X$,