# Helly and Radon theorems for convex intersections containing k-flats

Authors: Sarah Ludwigson; Pablo Soberón
Year: 2026
Status: preprint
Paper page: https://www.psoberon.com/papers/helly-radon-flats/
arXiv: https://arxiv.org/abs/2608.27189

Topics: Helly-type theorems; Tverberg theory; Topological combinatorics
Student coauthors: Sarah Ludwigson

## Abstract

We study versions of results in combinatorial geometry related to families of convex sets in ℝᵈ whose intersection contains a k-dimensional affine space. We prove generalizations of the colorful Radon theorem, the fractional Helly theorem, the colorful Helly theorem, and the selection-structure Helly theorem. When k=0, our arguments give new proofs of the corresponding versions for points.

## Overview

Colorful and fractional intersection theorems for convex families whose intersection contains an affine k-flat.