We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius R such that, for any pairwise disjoint k-element subsets \(Q_1,\dots ,Q_n\) of a normed space, there exists a partition of \(Q_1\cup \cdots \cup Q_n\) into disjoint transversals \(\{P_1,\dots ,P_k\}\) for which a ball of radius R intersects the convex hull of each \(P_i\) ( \(1\le i\le k\) ). Our methods are deterministic and dimension-free, and they are unified by optimizing two functionals: a quadratic selection functional whose local maximizers produce a complete system of disjoint transversals, and a convex intersection functional that certifies a common point. First, in the Euclidean setting we bound R in terms of the Chebyshev radii (minimal enclosing-ball radii) of the color classes \(Q_1,\dots ,Q_n\) . A key observation is a “combinatorial” subadditivity of the squared Chebyshev radius: given sequences \(X=(x_1,\dots ,x_k)\) and \(Y=(y_1,\dots ,y_k)\) of points in a Euclidean space, contained in balls of radii \(R_X\) and \(R_Y\) (not necessarily with the same center), one can reenumerate Y so that the pointwise-sum sequence \(Z=(x_1+y_1,\dots ,x_k+y_k)\) is contained in a ball of radius \(R_Z\) satisfying \( R_Z^2 \le R_X^2 + R_Y^2 . \) As a corollary, we obtain the best-possible bound \( R \le \frac{1}{\sqrt{2n}}\sqrt{\frac{k-1}{k}}\, \max _{1\le i\le n} \operatorname {diam}(Q_i). \) Our algorithm returns the desired disjoint transversals in overall time \(\mathcal {O}(nk^3)\) . Second, we develop a complementary approach based on the inter-color diameter and extend the framework to obtain no-dimensional colorful Tverberg-type results in the hyperbolic setting and in Banach spaces.
