17 problems

Continuity conjecture. For every stationary ergodic process $Z$,

Madiman–Kontoyannis conjecture. The entropy satisfies

Strengthened varentropy conjecture. The function $K$ is log-concave: $\log K(t)$ is concave on $(-\gamma,+\infty)$.

Let $h_\alpha$ denote the binary conditional Rényi-entropy function and let $\ast$ denote binary convolution. There exists a value $1<\hat\alpha<1.3863$ such that … is convex for…

Let $k_\alpha^A$ denote the Arimoto conditional-entropy function and let $\ast$ denote binary convolution. There exists a value $1<\hat\alpha<1.5783$ such that … is convex for…

For $\beta>0$, let the generalized Gaussian have density … where $\beta$ and $\alpha$ are chosen so that it integrates to one. Let $X_j$, $j=1,\dots,n$, be independent random variab…

Cross-section body conjecture. If $s\in[-1/n,\infty]$, then $C_p(f)$ is a symmetric convex body. This is the geometric reformulation of the reverse Rényi entropy power norm conject…

Let $(X,Y)$ be a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$, and suppose that $X$ and $Y$ have equal $p$-Rényi entropy. Convexity conjecture for Rényi entropy. The…

Recall that a probability measure on $\mathbb{R}^n$ is $\kappa$-concave if it satisfies the Brunn–Minkowski-type inequality associated with $\kappa$; a random vector is $\kappa$-co…

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…

For $p>\frac{d}{d+2}$, define $\beta_p$ by … Let $Z^{(p)}$ be a random vector drawn from $g_{\beta_p}$. Let $X_1,\ldots,X_n$ be independent random vectors in $\mathbb{R}^d$, and le…

Independence conjecture. If $\rho$ is optimal for $U$ and one dual pair $(\alpha,\beta)$, then $\rho$ is also optimal for every other dual pair.

Sharp two-summand Rényi entropy power conjecture.

Rényi entropy power conjecture.

Let $\alpha>1$ and let $n\in\mathbb{N}$. Write ${\rm H}_{\alpha}(\textbf{X}_{RSS})$, ${\rm H}_{\alpha}(\textbf{X}^*_{RSS})$, and ${\rm H}_{\alpha}(\textbf{X}_{SRS})$ for the Rényi…