HEXAGONbeta

Sharp fractional Helly bounds for common affine flats

Primarily AI-generated textHuman understanding: some partsmath.MG — Metric Geometrymath.CO — Combinatorics

Contributed by Matěj Kripner ↗, Václav Rozhoň ↗, Robert Šámal ↗, Martin Šindelář, Adrián Zámečník

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

If an α\alpha fraction of the (d−k+1)(d-k+1)-tuples of closed convex sets in Rd\mathbb R^d have an intersection containing a whole affine kk-flat, then a fraction 1−(1−α)1/(d−k+1)1-(1-\alpha)^{1/(d-k+1)} of the sets share such a flat. This coefficient is best possible, resolving the sharp fractional De Santis question of Ludwigson and Soberón. The geometric result behind the bound is (d−k)(d-k)-collapsibility of the common-flat complex, proved by coordinating convex sweeps in varying quotient spaces.

Provenance statement

This work was produced within the Bolzano project (https://bolzano.app), using AI systems for mathematical research and manuscript preparation. The listed authors contributed to the project and the preparation of this release. The manuscript received only limited human checking.

Tools used

Bolzano team
Bolzano

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Version history

  1. v1Submitted by Václav RozhoňInitial depositCurrentOct 15, 2026