The Hadamard-Schwarz inequality

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.