Year: 2025
Type: article
Abstract: In this paper, we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces ( P p ( R d ) , W p ) (\mathcal{P}_{p}(\mathbf{R}^{d}),W_{p}) . When d = 1 d=1 , as ( P 2 ( R ) , W 2 ) (\mathcal{P}_{2}(\mathbf{R}),W_{2}) is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When d > 1 d>1 , we show the failure of this property in two regimes: when p > 1 p>1 is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces ( P p ( R d ) , W p ) (\mathcal{P}_{p}(\mathbf{R}^{d}),W_{p}) when d ≥ 2 d\geq 2 and p ∈ ( 1 , + ∞ ) p\in(1,+\infty) are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion . As a corollary of the former, they have nonnegative upper Alexandrov curvature , in a precise sense that
Authors: Anshul Adve, Alpár R. Mészáros
Institutions: Princeton University, Durham University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414318781