Milman–Szarek geometric lemma on vertices and covering numbers

Establish the Milman–Szarek geometric lemma: For any r in (0, 1], prove that there exists a constant c = c(r) such that for every dimension n in N and every polytope P = conv(V) subset of R^n containing r B_2, if the Euclidean covering number satisfies N(P, B_2) ≤ e^{c n}, then the number of vertices satisfies |V| ≥ e^{c n}.

Background

The paper investigates relationships between geometric and combinatorial properties of high-dimensional polytopes, focusing on covering numbers by the Euclidean unit ball. Milman and Szarek formulated a conjectural geometric lemma connecting the covering number N(P, B_2) and the number of vertices |V| of a polytope P that contains a scaled Euclidean ball r B_2.

While the duality conjecture for metric entropy has been proved, the original geometric lemma by Milman and Szarek remains unresolved. This work proves an analogue of the conjectured statement for facets rather than vertices, highlighting that a duality argument transforming facet bounds into vertex bounds is not known to the authors. The open problem is to prove the original vertex-based geometric lemma.

References

Conjecture (Milman-Szarek). Let r ∈ (0,1]. Then there exists c = c(r) such that for every n ∈ N and for every polytope P = conv(V) ⊂ Rn containing r Bn, then N(P, B_2) ≤ e{c n} ⇒ |V| ≥ e{c n}. In particular, the covering number N and number of vertices |V| cannot both be sub exponential (in the dimension). Milman and Szarek showed that this conjecture implies the "duality conjecture" regarding covering numbers, and while the latter has since been proved (see [2]), the former is still open.

— On the Many Faces of Easily Covered Polytopes (2410.17811 - Florentin et al., 23 Oct 2024) in Section 1 (Introduction and the main result)