Open question · OPEN-E-SPECTRAL-GAP-BOUND

Does a spectral gap control E geometry?

Openpublication: open

Meaning

Seek a quantitative finite necessity bound under eigenvalue separation.

Statement

Under a simple minimum eigenvalue with gap gamma, determine whether exchange stability implies an approximate rank-one Voronoi rule with explicit w/gamma scaling.

Relationships

Related settled results

E-optimality supergradient structureE-SUPERGRADIENT
Positive first-order E margin need not imply exact finite improvementE-FIRSTORDER-NOT-FINITE

Where it comes from

Raised by

Positive first-order E margin need not imply exact finite improvementE-FIRSTORDER-NOT-FINITE
E-optimality supergradient structureE-SUPERGRADIENT

Local graph

Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Quantitative E necessity bound under a spectral gap

For a simple $\lambda_{\min}$ separated by $\gamma>0$ , does exchange stability imply an approximate rank-one Voronoi rule with an explicit $O(w/\gamma)$ bound? Population companion: OPEN-E-COMMON-SUPERGRADIENT (KNOWN_RESULTS/06-e-optimality.md § E6).

From OPEN_PROBLEMS.md