In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound $$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol} \frac{\stsys_2^n}{\vol}\leq n! $$ where $\stsys_2$ is the stable 2-systole, defined as the least stable norm of a non-torsion 2-homology class (in $H_2(M;\mathbb Z)$). Here the stable norm is the infimum of areas of representative rational 2-cycles. The inequality is tight since the Fubini-Study metric attains the boundary case of equality.
Recently we obtained a natural extension of the form $$ \frac{\stsys_2^n}{\vol}\leq n!\, (\Gamma_b)^n $$ for 2-essential manifolds $M$ with Betti number $b=b_2(M)$, where $\Gamma_b$ is the maximal product of successive minima $\lambda_1(L)\lambda_b(L^\ast)$ for $b$-dimensional lattices $L$ (so that $\Gamma_1=1$).
Since the inequality gets weaker for large $b$, one is led to ask whether the constant could be replaced by a universal constant depending only on the dimension of the manifold?
This is true for $1$-systoles of $1$-essential manifolds (also due to Gromov, based on the inquality discussed at On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$ ). One can certainly ask a more general question about the higher stable systoles, but the case of the stable 2-systole seems interesting enough.
A related article appeared here.
