A bound for the exterior product of S-units

We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of S-units contained in a number field k. This leads to a bound for the exterior product of S-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the S-unit group but not on the field k. Our inequality is related to a conjecture of F. Rodriguez Villegas.