Given a finite set $P$ of $n\gt d+1$ points in $d$-dimensional euclidean space.
Under the assumption that the points of $P$ are in general position in the sense that $\lbrace p_{i_1},\dots,p_{i_d}\rbrace\subset P\implies \alpha_{i_1}p_{i_1}+\cdots+\alpha_{i_d}p_{i_d}\not\in P$:
Question:
is it true that $p_{i_0}\in P$ is inside $P$'s convex hull $CH(P)$, i.e. can be expressed as $p_{i_0}=\sum\limits_{j=1}^{d+1}\alpha_{i_j}p_{i_j},\quad \lbrace p_{i_1},\dots,p_{i_{d+1}}\rbrace\subset P,\,0\lt\alpha_{i_j}\lt 1 \\ \iff \exists\ \lbrace q,t_1,\dots,t_d\rbrace\subset P:\quad(q-p)^T(t_i-p)\lt0\quad\forall i\in\lbrace 1,\dots,d\rbrace$
If that is indeed the case it would be possible find the corners of convex hulls from length measurements alone, making a generalization to arbitrarily weighted complete symmetric graphs possible.
The essential benefit would be having replaced the antisymmetric outer product with the symmetric inner product of vectors.
Another aspect would be that identifying the corners of convex hulls in $d$ dimensional euclidean space would be possible $O(n^3)$ time by iterating over $n$ points as candidates in the role of $p$, $n-1$ further points in the role of $q$ and the remaining $n-2$ points for which the number of negative inner products are counted.
