The Hadamard‐Schwarz inequality

Given α 1 , …, α k arbitrary exterior forms in of degree l 1 , …, l k , does it follow that | α 1 ∧⋯∧ α k | ≤ | α 1 | ⋯ | α k | 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 C n independent of the forms satisfying | α 1 ∧⋯∧ α k | ≤ C n | α 1 | ⋯ | α k | 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 l 1 , …, l k 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 ≤ l 1 , …, l k ≤ n .