6 problems

Let $X$ be a Hadamard space, and let its asymptotic rank be the largest dimension of a Euclidean flat that occurs asymptotically. An isoperimetric filling inequality is a bound on…

Gromov's conjecture. Every proper cocompact Hadamard space admits a linear isoperimetric inequality for $\mathbf I_k(X)$ whenever $k\geq\operatorname{Rank} X$. This conjecture pred…

Let $Y$ be the object under consideration, let $p$ be a point, and let $TC_pY$ denote its tangent cone at $p$. Write $0_p$ for the zero element in this tangent cone, and let…

Let the setting be that of the preceding result, with transformed empirical Fréchet means and a steep variational inequality of exponent $\beta$. The steep variational inequality i…

A Hadamard space is a complete geodesic metric space of nonpositive curvature in the CAT(0) sense. Consider the entire hierarchy of quadratic metric inequalities obtained in the pa…

Heat-flow coincidence conjecture. The heat flow semigroup $(T_t)$ defined by the paper's formula for $F:=P_{(D)}$ coincides with the heat flow constructed by Sturm.