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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

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