The Hadamard-Schwarz inequality. (Q1880550)

Summary: Given \(\alpha^1,\dots, \alpha^k\) arbitrary exterior forms in \(\mathbb{R}^n\) of degree \(l_1,\dots,l_k\), does it follow that \[ | \alpha^1 \wedge\dots\wedge \alpha^k| \leq|\alpha^1 |\cdots | \alpha^k|? \tag{1} \] 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 \[ |\alpha^1 \wedge \dots\wedge \alpha^k|\leq C_n| \alpha^1| \cdots |\alpha^k|. \tag{2} \] 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 \(l_1,\dots, l_k\) 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 \(0\leq l_1,\dots,l_k\leq n\).