Quick answer: "Triangle" becomes "tetrahedron" in 3 dimensions, and "simplex" in any number of dimensions. There is a notion of "solid angle" that works in any number of dimensions. Then the sum of distances to the vertices of a simplex is minimized (1) by a point inside it from which the solid angles subtended by the faces of the simplex are equal, or (2) at one vertex of the simplex; case 2 holds if the angle subtended at some vertex is >= 1/(n+1) of the "total solid angle" in that many dimensions, otherwise case 1 does.

I suppose I should explain what all that means.

Simplices

A simplex in n-dimensional space consists of n+1 points in general position, together with every convex combination of those points.

"In general position" means they don't all lie in a single hyperplane.

A "hyperplane" means all the points satisfying some linear relationship (strictly, affine rather than linear) between their coordinates. E.g., ax+by+c=0 in two dimensions.

A "convex combination" of x1,...,xk means all points of the form a1.x1 + ... + ak.xk where a1,...,ak are non-negative numbers whose sum is 1. E.g., the convex combinations of two points are exactly the points on the line segment joining those two points.

Solid angles

Suppose you have a point P in n-dimensional space, and some other convex region R in the space. Consider all the rays from P that pass through R; they form a "convex cone" based at that point.

A "convex region" is one such that, if a bunch of points belong to it, so do all convex combinations of those points.

Now, take a (hyper)sphere of radius 1 centred at P and look at the points where those rays meet its surface. The (hyper)area of the set of such points is the solid angle subtended at P by R.

The solid angle subtended at any point by the whole of space equals the (hyper)area of the unit sphere in that space. There is a formula for this but it doesn't matter right now. (When n=2 the "unit hypersphere" is actually a unit circle and "hyperarea" actually means "length"; the figure is 2pi, so we are working in radians. When n=3 it's a unit sphere, "hyperarea" means area, and it's 4pi, so the maximum solid angle you can have is 4pi.)