On nonexpansiveness of metric projection operators on Wasserstein spaces

In this paper, we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">P</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="bold">R</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi>W</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Pₚ(Rᵈ),Wₚ) . When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> d=1 , as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">P</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="bold">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi>W</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (P₂(R),W₂) 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> d>1 , we show the failure of this property in two regimes: when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> p>1 is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">P</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="bold">R</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi>W</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Pₚ(Rᵈ),Wₚ) when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> d≥ 2 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> p∈(1,+∞) 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 we define here. In our analysis, a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.