We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may...

Motivated by Kloeckner's result on the isometry group of quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$, we describe $\mathrm{Isom}\left(\mathcal{W}_p (E)\right)$ for all parameters $0 1$, solving problem affirmatively existence mass-splitting isometries.

The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space a...