Geometric characterizations of centroids of simplices
The authors study centroids of different \(k\)-skeletons of simplices in Euclidean \(d\)-space assuming constant density. They prove that the centroid of the full-dimensional simplex \(S\) and of its vertex set coincide, and that the centroids of the other skeleta are generically different. In addition, types of simplices are investigated for which all the centroids for all values of \(k\) coincide. E.g., it is shown that for any \(d= pq-1\), where \(p\) and \(q\) are integers larger than 2, there exists a non-regular \(d\)-simplex for which the centroids of all \(k\)-skeletons, \(1\leq k\leq d\), coincide, and the conjecture is made that the same holds if \(d+1\) is prime and larger than 3. On the other hand it is proved that the centroids for all \(k\) with \(0\leq k\leq d\) coincide iff the sum of the \(k\)-dimensional areas of the \(k\)-faces containing each specific vertex equals \((k+1)/(d+1)\) times the total \(k\)-dimensional area of the \(k\)-skeleton of \(S\). This is applied to verify the existence of a non-regular 4-simplex all whose centroids coincide. Further on, examples are presented for which centroids coincide in pairs, and it is also shown that the centroids do never coincide in triples if the polytopes under consideration are regular pyramids.
scientific article; zbMATH DE number 5016169