Flat subsets of Euclidean buildings

TL;DR

This paper proves that convex flat subsets in complete Euclidean buildings are always contained within a single apartment, clarifying the structure of such subsets in these geometric spaces.

Contribution

It establishes a fundamental property of convex flats in Euclidean buildings, linking them to maximal apartments, which was previously unknown.

Findings

Convex flats are contained in a single apartment.

Provides structural insight into Euclidean buildings.

Enhances understanding of geometric properties of Euclidean buildings.

Abstract

We prove that any convex flat subset in a complete Euclidean building is contained in an apartment of the maximal system of apartments.