Optimality of no-dimensional bounds in Banach spaces

We discuss lower-bound constructions for several no-dimensional theorems of combinatorial geometry in Banach spaces. The common mechanism is the Maurey--Pisier theorem: the supremal Rademacher type of a Banach space forces finite-dimensional $\ell_p$-structures, and standard-coordinate configurations in these model spaces give lower bounds for the error terms. For the Helly approximation property the relevant type is the type of the dual space. For colorful Radon, colorful Tverberg, selection, and weak $\varepsilon$-net statements the relevant type is the type of the original space. We show that the powers appearing in the no-dimensional Helly, Radon, Tverberg, and selection estimates are optimal at the supremal-type exponent. If the supremal type is attained, the known upper estimates coming from the corresponding type inequalities have the best possible order. We also include endpoint statements for spaces of trivial type. In this case the error terms in the Helly, Radon, Tverberg, and selection statements cannot tend to zero. Finally, we prove an endpoint obstruction for no-dimensional weak $\eps$-nets in spaces of trivial type. For every fixed cardinality bound, one can find a finite set in the unit ball for which no approximate weak $\eps$-net of that size exists below a fixed positive radius. The proof combines the simplex example in $\ell_1^N$, the Lov\'asz theorem on the chromatic number of Kneser's graph, and finite representability of $\ell_1^N$ in spaces of trivial type.