Determining if a point in 3-space is inside a polytope knowing only the distances to the polytope's vertices

Question

If I have a point in 3-space, as well as a convex 3-polytope, and an unordered set of distances to the vertices of the 3-polytope (but not the position of these vertices) is there any way for me to determine if the point is inside the polytope? If not, what if we restrict the polytope to be a 3-simplex?

Answer 1

No, not even for a 3-simplex. Take our point to be the origin, and let $a<b<c<d$ be the distances from the origin to vertices of your 3-simplex. We will construct two polytopes with the specified distances, one with the origin inside the polytope, and one with the origin outside the polytope. Let $\varepsilon>0$ be sufficiently small.

• Let $A$ be a point on the line from the origin in the direction $(1,0,\pm \varepsilon)$ of length $a$
• Let $B$ be a point on the line from the origin in the direction $(-1,1,\pm\varepsilon)$ of length $b$
• Let $C$ be a point on the line from the origin in the direction $(-1,-1,\pm\varepsilon)$ of length $c$
• Let $D$ be a point on the line from the origin in the direction $(0,0,1)$ of length $d$.

Now, depending on if we choose the signs $\pm$ to all be positive or all negative, the origin is either outisde or inside the polytope which is the convex hull of $A,B,C,D$.