A pleasant exercise

TL;DR

This paper provides a proof for a combinatorial lemma mentioned by Langlands in 1965, which was previously unproven in the literature, contributing to the historical understanding of the subject.

Contribution

The authors supply a proof for Langlands' combinatorial lemma, clarifying an unproven statement from the 1965 AMS Boulder conference.

Findings

Proof of Langlands' combinatorial lemma provided

Clarification of historical mathematical statements

Enhancement of understanding of characteristic functions in Euclidean spaces

Abstract

In the Proceedings of the AMS Boulder conference in 1965 Langlands states a combinatorial lemma involving families of characteristic functions attached to ordered partitions of an obtuse basis in a finite dimensional euclidean vector space. Langlands does not give any indication about the proof of the lemma which is said to be a "pleasant exercise". Since we did not find a proof in the literature we decided to give one. We believe it of some interest for the history of the subject.