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Given $m≥d+1$ a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that

1) they all share the common vertex M

2) the simplices $\Delta_i$ triangulate the polytope $P=\mathrm{Conv}(\{V_{i,j}\}_{i=1,…,m|j=1,…,d}) $containing M in its interior?

If the answer is positive, is there a simple algorithm to produce an instance of the points V given m and d?

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You essentially ask for a $d$-dimensional polytope with $m$ simplicial facets. Already for $d=3$ you have a problem with $m$ odd, as the number of edges is $3m/2$, because each edge lies in exactly 2 triangles.

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  • $\begingroup$ Note that the essential bit is "containing $M$ in its $interior$". - Else, at least as long as the simplices are not bound to be regular, and for realizations of abstract polytopes I assume so, the above argument simply would become wrong. Consider the $m$-gonal pyramid, where $M$ happens to be the center of the polygonal base, and all other vertices are your $V_i$. Then you clearly can have an adjoin from any $m\ge d=3$ (both odd $and$ even, and esp. $m=3$ too!). - As $M$ here happens to be on the boundary, several "edges" do not count in the to be considered hull $P$. $\endgroup$
    – Dr. Richard Klitzing
    Commented Nov 15, 2019 at 19:17

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