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.

This question is related to [this post][1].


  [1]: https://mathoverflow.net/questions/397420/holder-continuous-barycenter-maps