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Question:
given $k,\,k>n$ points in convex configuration and general position in $n$ dimensional Euclidean space, i.e. no $n+1$ points of which are co-hyperplanar,

what can be said about how the sum of the volumes of the set of simplices defined by the subsets of $n+1$ points depends on the volume of the convex polytope defined by all $k$ points?

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  • $\begingroup$ $V(hull)=\sum_i V(simplex_i)$ if $\{simplex_i\}$ eg. form Delone "cells", i.e. you would exclude to count internal points by multiple different simplices. $\endgroup$ Commented Jan 1 at 10:44
  • $\begingroup$ @Dr.RichardKlitzing not sure if I understand your comment right, but in the case of 2D strictly convex polygons we have for 3 corners that the volume of the polygon equals the sum over all simplex volumes; in case of convx quadrilateral the sum over all simplex volumes is twice the polgon volume; what e.g. about strictly convex polygons in 2D with 5,6,... corners; what about strictly convex polyhedra in higher dimensions? $\endgroup$ Commented Jan 1 at 13:09
  • $\begingroup$ that's why I asked to exclude multiple coverages, i.e. using a single dissection into a Delone set of triangles (simplices). Cf. en.m.wikipedia.org/wiki/Delone_set $\endgroup$ Commented Jan 1 at 18:20

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