Let $x_0,\ldots, x_n$ be $(n+1)$ points in $\mathbb{R}^n$ in general position and let $d_{ij}=d(x_i,x_j)$ for all $0\leq i<j\leq n$ be their pairwise distances. Question: Is there a condition on the distances $d_{ij}$ under which the center of the circumsphere of the $n$-simplex spanned by the points $x_1,\ldots, x_n$ lies inside the simplex ?
For example, in the case of a triangle with edge lengths $a,b,c$, the center of the circumcircle lies inside the triangle if and only if the triangle is acute. The triangle is acute if and only if $a^2\leq b^2+c^2 \land b^2\leq a^2+c^2 \land c^2\leq a^2+b^2$. It lies on the boundary if there is some equality.
If the general case is too complicated what about the tetrahedron ? Given a tetrahedron with edge lengths $a,b,c,d,e,f$. Are there inequalities analogous to the ones for a triangle defining an acute tetrahedron ?