Question:Is it true that $E^2$ is the only Euclidean space, in which the convex hull of $n+2$ points in convex configuration has two inner diagonals and in all other cases there is only one such diagonal?

In this context an inner diagonal shall be a line-segment that connects two corners of the convex hull and, for which each inner point can be expressed as the convex-combination of *all* corners and all weights in the open unit interval,

i.e. if the inner points of the segment connecting corners $p_i$ and $p_j$ can be expressed as $\ \alpha p_i+(1-\alpha)p_j\ $ and, as $\ \sum_{k=1}^{n+2}\beta_kp_k$, $\quad0\lt \alpha, \beta_k\lt 1$

In $E^1$ maximally $2$ points can be in convex configuration, but the linesegment connecting the two points complies with the definition of inner diagonals.