# Estimation of the Gromov–Wasserstein distance of spheres

Let $$(X,d_X,\mu_X)$$ and $$(Y,d_Y,\mu_Y)$$ be two metric measure spaces. A probability measure $$\mu$$ over $$X\times Y$$ is called a coupling if $$(\pi_1)_\sharp \mu=\mu_X$$ and $$(\pi_2)_\sharp \mu=\mu_Y$$. We define the Gromov–Wasserstein distance $$d_{\mathcal GW}(X,Y)$$ $$^{[1]}$$ by

$$\frac{1}{2}\inf_\mu(\int \int |d_X(x,x')-d_Y(y,y')|^p\mu(dx\times dy)\mu(dx' \times dy'))^{1/p}$$

where the infimum is taken over all the couplings of $$\mu_X$$ and $$\mu_Y$$.

I wonder if there are any computations/estimate for $$d_{\mathcal GW}(S^m,S^n)$$, where the distances are geodesic distances and measures are uniform measures on spheres.

Reference: [1] Mémoli, F. Found Comput Math (2011) 11: 417. https://doi.org/10.1007/s10208-011-9093-5

I calculated quite a few examples using a quantization method to discretize the uniform distributions on the spheres. Numerical results suggest that the following coupling is optimal (for all $$p\geq 1$$), even though I cannot prove it:
Let $$m > n$$ and $$\mu_X$$ be the uniform distribution on $$S_m$$ and $$\mu_Y$$ the uniform distribution on $$S_n$$. Let $$\mu = \mu_X \circ (id, T)^{-1}$$, where $$T$$ is simply a projection of the spherical coordinates. So $$(\varphi_1, ..., \varphi_{m-1})$$ is mapped to $$(\varphi_{m-n+1}, ..., \varphi_{m-1})$$.
Notably, it is not even clear to me that the second marginal of $$\mu$$ again gives the uniform distribution on $$S_n$$, even though I guess people more familiar with spherical coordinates might see this immediately.