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Is it possible to construct convex D-polyhedron $P$ that there exists a D-simplex $A$ spanned on vertices of $P$, that its volume $V_A$ only decrease if any of its vertices is shifted into a neighbouring vertex of $P$, but there is another D-simplex $B$ spanned on vertices of $P$ the volume of which $V_B$ is greater then volume of $A$?

Another words can I iteratively find maximal simplex enclosed by convex hull of a set of points or are there local maxima?

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