On the sum of the angles between three vectors

Iosif Pinelis

Home/Archive/arXiv

arXiv · 2508.04759

On the sum of the angles between three vectors

Iosif Pinelis

View original source ↗Read source document ↗Export metadata ↓

Abstract

For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2π$. As an easy consequence of this, a proof of Euclid's theorem that the sum of the interior angles of any triangle is $π$ is provided. So, the main result of this note can be considered a generalization of Euclid's theorem. To a large extent, the consideration is reduced almost immediately to a choice for the sum of three related angles among the three integer multiples $0,2π,4π$ of $π$. The rest of the consideration concerns only various betweenness relations.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Iosif Pinelis. 2025-09-21. On the sum of the angles between three vectors. https://arxiv.org/abs/2508.04759

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗

arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗

arXiv2026

Minimal central slices of the regular simplex

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

math.MG↗