The ACO Seminar (2026–2027)

In this talk, I will discuss a tight colorful dimension-free Tverberg theorem. If $P_1,\dots, P_k$ are finite disjoint sets of $n$ points in a Euclidean space, then our result guarantees a partition of the union $P_1 \cup\dots\cup P_k$ into $n$ rainbow sets of size $k$ such that a single ball intersects the convex hull of every set. The radius of this ball depends only on the number of color classes, the number of rainbow sets, and the largest diameter of a color class --- not on the dimension of the ambient space. Moreover, the resulting bound is best possible.

The key tool is a Pythagorean-type subadditivity property of the Chebyshev radius, which is of independent interest. Given sequences $X=(x_1, \dots, x_n)$ and $Y=(y_1, \dots, y_n)$ of points in a Euclidean space, contained in balls of radii $R_X$ and $R_Y$, respectively, one can re-enumerate $Y$ so that the pointwise-sum sequence $Z=X+Y$ is contained in a ball of radius satisfying $R_Z^2 \leq R_X^2+R_Y^2$.

This is joint work with Polina Barabanshchikova and Grigory Ivanov.