The Covering Polarity

Cover a convex body with small balls. How many do you need? Now take the polar dual of the body and cover that. Classical covering duality — the König-Milman result — relates these two numbers when the covering uses translates of a fixed set. The metric is translation-invariant, and the duality is clean.

Hilbert geometry breaks translation-invariance. The Hilbert metric on a convex body is defined intrinsically — distances depend on where you are within the body, not just on the displacement. Covering a convex body with Hilbert balls is a fundamentally different problem from covering it with translates.

Vernicos et al. prove that polarity duality still works. There exists an absolute constant such that the covering numbers of a body and its polar dual are polynomially related, even without translation-invariance. They recover the classical volumetric duality as a special case and establish a new boundary-covering duality connecting interior coverings to boundary coverings.

The technical challenge is controlling how polarity interacts with the Hilbert metric’s position-dependence. The authors develop α-expansions and stability lemmas that quantify this interaction, combined with localized isoperimetric arguments and Holmes-Thompson area estimates. The machinery is heavy because the problem is genuinely harder: without translation-invariance, every ball in the covering has a different shape, and the polarity map distorts them all differently.

The result says the duality is robust — it survives the loss of the symmetry that made it discoverable.