Polynomial-time algorithm for Adiprasito–et al. dimension-free bound in ℓ^2

Develop a polynomial-time algorithm that produces the partition guaranteed by Adiprasito, Bárány, and Soberón’s averaging-argument bound for the colorful no-dimension Tverberg problem in ℓ^2, i.e., computes k disjoint transversals P_1, ..., P_k achieving R(ℓ^2, k, r) ≤ (1+√2)/√r with the prescribed intersection property.

Background

Adiprasito et al. proved a dimension-free colorful Tverberg-type bound in the Euclidean setting using an averaging argument, establishing R(ℓ^2, k, r) ≤ (1+√2)/√r. However, their proof does not yield a known polynomial-time algorithm to construct the required partition, leaving an algorithmic gap.

Subsequent work by Choudhary and Mulzer gave a polynomial algorithm via Sarkaria’s tensor trick, but it achieves a weaker bound O(√(k/r)) and thus does not match the Adiprasito–et al. guarantee. The present paper provides tight bounds and polynomial algorithms for different formulations, but the specific algorithmic realization of the Adiprasito–et al. bound remains unresolved.