A note on E. Thiémard's algorithm to compute bounds for the star discrepancy

Thiémard (J. Complexity 17(4) (2001) 850) suspects that his upper bound for the discrepancy B(P,x⇒) is bounded below by a measure for the tightness of the partition P. We present a counterexample but also prove that for a certain class of partitions the assumption is valid. The partitions used in Thiémard's algorithm are all part of this class.