11 problems

Let $L$ denote the set of all real random variables $X$ with a log-concave probability density function such that $EX=0$ and $EX^2=1$. Let $Y$ have the standard exponential distrib…

Let $f_X(x)=e^{-\phi(x)}$ be a symmetric log-concave distribution, where $\phi$ is upper bounded by some fixed polynomial. Consider one-bit non-adaptive and adaptive estimators of…

For the generalized Gaussian family with shape parameter $\beta>0$ and unit-variance density … where the distribution is strictly log-concave for $\beta>1$, the non-adaptive lower-…

Consider the random surface models on the $d$-dimensional bipartite torus and box, with zero boundary values, sampled from the corresponding surface measure with interaction potent…

The paper considers a distribution of the form $p(x)\propto e^{-f(x)}$, where $f$ is close to a mixture of log-concave functions in the sense that there exists…

Exponential extremizer conjecture. The extremal distribution for this problem is the exponential distribution.

Let a probability density function be log-concave, and consider the constrained Lloyd–Max quantization algorithm with a minimal cell-probability constraint, referred to as Algorith…

Let $X_1$ and $X_2$ be log-concave random variables, and consider the entropy of their projection in a unit direction in $\bb{R}^2$. Ball et al.'s conjecture. The entropy-maximizin…

Optimal sample complexity conjecture. The optimal sample complexity for estimating log-concave densities is

Variance conjecture. Every log-concave isotropic random vector $X$ satisfies

Let $X_i$ be iid random vectors from a normalized isotropic log-concave distribution. Universality conjecture. The results of Theorem hold for these random vectors. This conjecture…