Note on the degree of concavity of the determinant on sets of positive definite quadratic forms

An upper bound is determined for the for whichD(f) is a concave function off, wheref ranges over Minkowski-reduced positive definite quadratic forms inn variables with diagonal coefficients unity andD(f) denotes the determinant off In answer to a question of C E Nelson, it is shown thatD(f) is not concave in the casen 4