The Fundamental Theorems of Affine and Projective Geometry Revisited

Shiri Artstein-Avidan; Boaz A. Slomka

Home/Archive/arXiv

arXiv · 1604.01762

The Fundamental Theorems of Affine and Projective Geometry Revisited

Shiri Artstein-Avidan·Boaz A. Slomka

View original source ↗Read source document ↗Export metadata ↓

Abstract

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n+2 fixed projective points in real n-dimensional projective space , through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Shiri Artstein-Avidan, Boaz A. Slomka. 2016-04-06. The Fundamental Theorems of Affine and Projective Geometry Revisited. https://arxiv.org/abs/1604.01762

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

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗