6 problems

Let $d \ge 2$, let $\mu_i \preceq_c \nu_i$ for $i=1,\ldots,d$, and let $\vec\mu=(\mu_1,\ldots,\mu_d)$. A monotone coupling of $\vec\mu$ is the unique monotone probability measure w…

Let $\lambda_1$ and $\lambda_2$ be probability measures in convex order, let $\mathfrak{I}$ denote the interval supporting the relevant variables, and let $\mathcal{M}$ be the set…

Let $\mu$ and $\nu$ be the probability measures used in the one-dimensional martingale optimal transport experiments, with martingale couplings between them. An optimal coupling is…

Euclidean distance power conjecture. The conclusion of Theorem holds for

Maximal support bound conjecture. The bound is reached for every cost that is second order complete at infinity at $x_0$; equivalently, $N_c(x_0)$ attains the universal upper bound…

Ghoussoub–Kim–Lim conjecture. In higher-dimensional martingale optimal transport for distance cost, the optimal plans should be supported on $d+1$ graphs.