  We conjecture that a convex polytope is uniquely determined up to isometry by
its edge-graph, edge lengths and the collection of distances of its vertices to
some arbitrary interior point, across all dimensions and all combinatorial
types. We conjecture even stronger that for two polytopes $P\subset\mathbb R^d$
and $Q\subset\mathbb R^e$ with the same edge-graph it is not possible that $Q$
has longer edges than $P$ while also having smaller vertex-point distances.
  We develop techniques to attack this question and verify it in three relevant
special cases: if $P$ and $Q$ are centrally symmetric, if $Q$ is a slight
perturbation of $P$, and if $P$ and $Q$ are combinatorially equivalent. In the
first two cases the statements stay true if we replace $Q$ by some graph
embedding $q\colon V(G_P)\to\mathbb R^e$ of the edge-graph $G_P$ of $P$, which
can be interpreted as local resp. universal rigidity of certain tensegrity
frameworks. We also establish that a polytope is uniquely determined up to
affine equivalence by its edge-graph, edge lengths and the Wachspress
coordinates of an arbitrary interior point.
  We close with a broad overview of related and subsequent questions.
