Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-x_i|^2$, $p\in E^n$, just because $I(p)=I(y)+|p-y|^2$. In particular, it can not happen that all distances $|y-x_i|$ are at least $r$, but for some other point $p$ all distances $|y-x_i|$ are less than $r$.

Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-x_i|^2$, $p\in E^n$, just because $I(p)=I(y)+|p-y|^2$. In particular, it can not happen that all distances $|y-x_i|$ are at least $r$, but for some other point $p$ all distances $|p-x_i|$ are less than $r$.

Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. It minimizes the moment of inertia $\sum m_i |p-x_i|^2$, $p\in E^n$. In particular, it can not happen that all distances $|y-x_i|$ are at least $r$, but for some other point $p$ all distances $|y-x_i|$ are less than $r$.

Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-x_i|^2$, $p\in E^n$, just because $I(p)=I(y)+|p-y|^2$. In particular, it can not happen that all distances $|y-x_i|$ are at least $r$, but for some other point $p$ all distances $|y-x_i|$ are less than $r$.