Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following optimality set non-empty: $$ X(\mathbb{P}):=\left\{ x\in \mathbb{R}^n:\, \int_{u \in \mathbb{R}^n}\|u-x\|^p=\inf_{x'\in \mathbb{R}^n} \int_{u \in \mathbb{R}^n}\|u-x'\|^p \right\}. $$

More importantly, for which such $p$ does there exist a continuous selector $S:\mathcal{P}_p(\mathbb{R}^n)\rightarrow 2^{\mathbb{R}^n}$ with: $$ S(\mathbb{P})\in X(\mathbb{P}) ? $$ Note, here, we equip $\mathcal{P}_p(\mathbb{R}^n)$ with the $L^p$-Wasserstein distance.

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Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following optimality set non-empty: $$ X(\mathbb{P}):=\left\{ x\in \mathbb{R}^n:\, \int_{u \in \mathbb{R}^n}\|u-x\|^p\mathbb{P}(du)=\inf_{x'\in \mathbb{R}^n} \int_{u \in \mathbb{R}^n}\|u-x'\|^p\mathbb{P}(du) \right\}. $$

More importantly, for which such $p$ does there exist a continuous selector $S:\mathcal{P}_p(\mathbb{R}^n)\rightarrow \mathbb{R}^n$ with: $$ S(\mathbb{P})\in X(\mathbb{P}) ? $$ Note, here, we equip $\mathcal{P}_p(\mathbb{R}^n)$ with the $L^p$-Wasserstein distance.

This question is related to this post.