On comass and stable systolic inequalities
James J. Hebda, Mikhail G. Katz
TL;DR
The paper investigates the maximal ratio $C_{n,p}$ of the Euclidean norm to the comass norm for $p$-covectors in $\mathbf{R}^n$ and presents improved upper bounds beyond the classical $C_{n,p} \le \binom{n}{p}^{1/2}$. It develops inductive and combinatorial bounds for $C_{n,p}$, extends wedge-product estimates in the spirit of GHK, and computes exact values for selected $(n,p)$ using calibrations. These norm bounds are then applied to derive stable systolic inequalities on manifolds where the fundamental cohomology class is a cup product of lower-degree forms, yielding explicit constants and dimension-specific results, such as in dimensions $6$, $7$, and $8$ with equality cases tied to canonical geometries like $\mathbf{CP}^m$. The work thus sharpens the connection between norm comparisons, wedge-product behavior, and geometric inequalities in systolic geometry, and clarifies how lattice constants (e.g., $\Gamma_b$ or $\gamma_b$) influence the sharpness of these bounds.
Abstract
We study the maximum ratio of the Euclidean norm to the comass norm of p-covectors in Euclidean n-space and improve the known upper bound found in the standard references by Whitney and Federer. We go on to prove stable systolic inequalities when the fundamental cohomology class of the manifold is a cup product of forms of lower degree.