Largest j-simplices in d-cubes: Some relatives of the Hadamard maximum determinant problem

This paper studies the computationally difficult problem of finding a largest j-dimensional simplex in a given d-dimensional cube. The case in which j = d is of special interest, for it is equivalent to the Hadamard maximum determinant problem; it has been solved for infinitely many values of d but not for d = 14. (The subcase in which j = d = 3 (mod 4) subsumes the famous problem on the existence of Hadamard matrices.) The known results for the case j = d are here summarized and used, but the main focus is on fixed small values of j. When j = 1, the problem is trivial, and when j = 2 or j = 3 it is here solved completely (i.e., for all d). Beyond that, the results are fragmentary but numerous, and they lead to several attractive conjectures. Some other problems involving simplices in cubes are mentioned, and the relationship of largest simplices to D-optimal weighing designs is discussed.