Prove that a metric space \((X, d)\) has strong negative type if and only if the space of probability measures on \((X, d)\) equipped with the energy distance \(D\) has strong negative type.

According to Schoenberg [1937] and Schoenberg [1938b] \((X, d)\) is of negative type iff there is a Hilbert space \(H\) and a map \(\Phi:(X, d) ightarrow H\) such that \(d\left(x, x^{\prime}ight)=\left\|\Phi(x)-\Phi\left(x^{\prime}ight)ight\|^{2}\). Now the barycenter map \(\beta=\beta_{\phi}\) that maps a probability measure \(P\) with finite expectation to \(\int \Phi(x) d P(x)\) is an isometry into the Hilbert space \(H\); since Hilbert space has strong negative type, so does the set of probabilities. For more details on barycentric maps, see Proposition 3.1 of Lyons [2013] which explicitly says that if \((X, d)\) has negative type as witnessed by the embedding \(\Phi\), then \((X, d)\) is of strong negative type iff the barycenter map \(\beta_{\Phi}\) is injective on the set of probability measures on \((X, d)\) with finite first moment. The other direction of the iff is trivial because of degenerate probability distributions (that take one single value with probability \(1)\).