Theoretical analysis reveals upper bounds on simplex volumes for point sets in higher dimensions, highlighting new geometric constraints for Heilbronn triangle problem generalizations.
We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed , any subset of of size contains points that span a simplex of volume at most , points whose convex hull has volume at most , points that span a ‐dimensional simplex of volume at most .
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Dmitrii Zakharov (2024) studied this question.
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