On nonexpansiveness of metric projection operators on Wasserstein spaces

Let p ∈ ( 1 , + ∞ ) and let Ω ⊆ R d be closed and convex with L d ⁢ ( Ω ) ≥ 1 . Then, for K = K p ⁢ ( Ω ) and for any μ ∈ P p ⁢ ( Ω ) , problem (1.1) has a unique solution P Ω p ⁢ [ μ ] . Moreover, there exists B ⊆ Ω (depending on μ , p , Ω ) Borel measurable such that P Ω p ⁢ [ μ ] = μ ac ⁢ 1 B + 1 Ω ∖ B , where μ ac stands for the absolutely continuous part of 𝜇 with respect to L d . Here we have used the notation 1 B for the indicator function of 𝐵. Furthermore, for a Borel set X ⊂ R d , 1 X is interpreted as a Borel measure by 1 X ( A ) : = L d ( X ∩ A ) , for any Borel set 𝐴.

In the case of p = 2 , the projection P Ω 2 behaves well with respect to interpolation along generalized geodesics. Let μ , ν 0 , ν 1 ∈ P 2 ⁢ ( Ω ) with 𝜇 absolutely continuous. Then there are optimal maps T 0 , T 1 which send 𝜇 to ν 0 , ν 1 respectively. For t ∈ [ 0 , 1 ] , define the generalized geodesic connecting ν 0 and ν 1 with respect to 𝜇 by ν t = ( T t ) ♯ ⁢ μ , where T t = ( 1 − t ) ⁢ T 0 + t ⁢ T 1 . In [14], using the displacement 1-convexity of W 2 2 ⁢ ( μ , ⋅ ) along generalized geodesics, i.e. for all t ∈ [ 0 , 1 ] ,

it was shown that P Ω 2 is locally 1 2 -Hölder continuous. Since this argument is relying on the “Hilbertian like” behavior of W 2 , it is unclear to us whether such reasoning could be carried through for p ≠ 2 .

Let Ω ⊆ R d be closed and convex such that L d ⁢ ( Ω ) ≥ 1 and p ∈ ( 1 , + ∞ ) . The subspace K p ⁢ ( Ω ) defined in (1.2) is closed and geodesically convex in ( P p ⁢ ( Ω ) , W p ) .

Proof

The closedness of K p ⁢ ( Ω ) is straightforward. Let μ 0 , μ 1 ∈ K p ⁢ ( Ω ) . To show that K p ⁢ ( Ω ) is geodesically convex, we will show that the constant speed geodesic [ 0 , 1 ] ∋ t ↦ μ t connecting μ 0 to μ 1 is absolutely continuous with a density bounded above by 1. This is a consequence of [19, Theorem 7.28, Proposition 7.29] and [21, Corollary 19.5], but for completeness, we provide a sketch of this proof.

First, by [9, Lemma 3.14], we have that μ t ≪ L d ↾ Ω for all t ∈ [ 0 , 1 ] . To show the upper bound on μ t , we rely on the interpolation inequality for the Jacobian determinants of optimal transport maps provided in [9, Theorem 3.13] (see also [5] for p = 2 ). Let ϕ : Ω → R be the unique 𝑐-concave Kantorovich potential in the optimal transport of μ 0 onto μ 1 , where c ⁢ ( x , y ) = 1 p ⁢ | x − y | p . Then, by [9, Theorem 3.4], T ( x ) : = x − | ∇ ϕ ( x ) | q − 2 ∇ ϕ ( x ) (where 1 / p + 1 / q = 1 ) is the unique optimal transport map between μ 0 and μ 1 . Moreover, T t ( x ) : = x − t | ∇ ϕ ( x ) | q − 2 ∇ ϕ ( x ) is the unique optimal transport map between μ 0 and μ t .

Let us denote by Ω id ⊂ Ω the set where 𝜙 is differentiable and ∇ ϕ = 0 . Then, reasoning as in [9], we know that there exists a set B ⊆ Ω ∖ Ω id of full measure such that 𝜙 is twice differentiable on 𝐵 with det ( D ⁢ T ⁢ ( x ) ) > 0 if x ∈ B . Then, by [9, Theorem 3.13], we have

We remark that, because our underlying space Ω is flat, the volume distortion coefficients present in the previous inequality (stated in [9] for general Finslerian manifolds) become 1.

By (A.1), if det ( D ⁢ T ⁢ ( x ) ) ≥ 1 , we have det ( D ⁢ T t ⁢ ( x ) ) ≥ 1 , while if det ( D ⁢ T ⁢ ( x ) ) ≤ 1 , then det ( D ⁢ T t ⁢ ( x ) ) ≥ det ( D ⁢ T ⁢ ( x ) ) . In conclusion, det ( D ⁢ T t ⁢ ( x ) ) ≥ min ⁡ { 1 , det ( D ⁢ T ⁢ ( x ) ) } .

Now, since μ t = ( T t ) ♯ ⁢ μ 0 , when restricted to the set 𝐵, the change of variable formula yields

When restricted to the relative complement of 𝐵, T t essentially is the identity map, where the upper bound is also clearly preserved. The result follows. ∎