For each n≥ 1, define S_n to be the least constant such that, for every n-dimensional complex Banach space X and every invertible operator T∈L(X), |det T|,lVert T^-1rVert≤ S_n lVert T rVert^n-1. Determine the exact asymptotic behavior of S_n as n→∞, in particular the limit lim_n→∞S_n/√n. The cited preprint claims that this limit exists and equals √e.

Schäffer’s matrix inequality problem

Schäffer’s problem asks for the sharp constant in a determinant–inverse-norm inequality for matrices. The associated conjecture that this constant is independent of dimension was disproved, but its precise asymptotic value has remained unsettled.

Known results

Schäffer established $\mathcal{S}(n)\leq\sqrt{en}$ and conjectured dimension-independent boundedness.
Gluskin, Meyer, and Pajor disproved that conjecture in the early 1990s.
Bourgain and Queffélec subsequently improved the lower bounds.
A 2017 construction gave explicit Toeplitz counterexamples with growth of order $\sqrt{n}$ , matching the known upper-bound order without determining the sharp constant.

August 2026 asymptotic-constant claim

The preprint “Schäffer's matrix inequality: the exact asymptotic constant” claims a matching asymptotic lower bound using Wiener-algebra and model-operator constructions. If correct, it determines the sharp asymptotic constant, but the claim awaits peer review and is not yet independently verified.

Current status (as of August 2026): The dimension-independent conjecture is refuted and matching-order lower bounds are established, while the newly claimed sharp asymptotic constant remains unverified and the exact finite-dimensional constant is not confirmed.

Schäffer’s matrix inequality problem

Progress summary

Primary source

Additional references

Solutions 0