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 Δi 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?

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?