Given α1,…,αk arbitrary exterior forms in Rn of degree l1,…,lk, does it follow that |α1∧⋯∧αk|≤|α1|⋯|αk| The answer is no in general. However, it is a persistent, popular and even published misconception that the answer is yes. Of course, a routine calculation reveals that there exists at least a constant Cn independent of the forms satisfying |α1∧⋯∧αk|≤Cn|α1|⋯|αk| For reasons mentioned in the introduction, we refer to this as the Hadamard-Schwarz inequality. However, what the best constant is, either overall or for the particular numbers l1,…,lk remains well short of clear.It is the objective of this paper to explicitly describe the smallest constant for the Hadamard-¬Schwarz inequality as well as to identify the associated forms for which equality occurs. We have answered these questions for a wide class of integers 0≤l1,…,lk≤n.
