4 problems

Klartag's conjecture. The localisation method should generalise to the multidimensional setting by employing optimal transport of vector measures.

Let $\mu$ be an absolutely continuous Borel $\mathbb{R}^m$-valued measure on $\mathbb{R}^n$ with $\mu(\mathbb{R}^n)=0$, and let $u\colon\mathbb{R}^n\to\mathbb{R}^m$ be a $1$-Lipsch…

Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$, absolutely continuous with respect to Lebesgue measure, and let $f\colon\mathbb{R}^n\to\mathbb{R}^m$ be integrable with…

Let an extremal tuple of measures be a solution of the vector weighted logarithmic energy problem considered in the paper, with external fields satisfying the stated admissibility…