# Beckman-Quarles-type theorems

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A fundamental theorem in Euclidean geometry is the following result of F.S. Beckman and D.A. Quarles [a1]. Let $k$ be a fixed positive real number and let $f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} }$ ($n \geq 2$) be a mapping satisfying $d ( f ( x ) ,f ( y ) ) = k$ for all $x,y \in \mathbf R ^ {n}$ with $d ( x,y ) = k$. (Here, $d ( x,y )$ denotes the Euclidean distance

$$d ( x,y ) = \sqrt {\sum _ {i = 1 } ^ { n } ( x _ {i} - y _ {i} ) ^ {2} }$$

of $x = ( x _ {1}, \dots, x _ {n} )$ and $y = ( y _ {1}, \dots, y _ {n} )$ in $\mathbf R ^ {n}$.) The mapping $x \rightarrow f ( x ) - f ( 0 )$ is then in $O ( n, \mathbf R )$.

It should be emphasized that no regularity assumption (like differentiability or continuity) is required in the theorem. For an analogue in hyperbolic geometry, see [a4] and [a5].

A distance space is a set $S \neq \emptyset$, a set $W$ and a mapping $d : {S \times S } \rightarrow W$. The element $d ( x,y )$ is called the distance of $x$ and $y$ (in this order). Beckman–Quarles-type theorems for distance spaces are, up to generalizations, statements claiming that a mapping $f : S \rightarrow S$ preserves all occurring distances $w$ if it preserves one single distance. Beckman–Quarles-type theorems belong to a class of statements called characterizations of geometrical mappings under mild hypotheses.

Let $D ( x,y ) = ( x _ {1} - y _ {1} ) ^ {2} + \dots + ( x _ {n - 1 } - y _ {n - 1 } ) ^ {2} - ( x _ {n} - y _ {n} ) ^ {2}$, for $x,y \in \mathbf R ^ {n}$. Let $k \neq0$ be a fixed real number and let $f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} }$ ($n \geq 2$) be a mapping satisfying $D ( f ( x ) ,f ( y ) ) = k$ for all $x,y \in \mathbf R ^ {n}$ with $D ( x,y ) = k$. Then $x \rightarrow f ( x ) - f ( 0 )$ is a Lorentz transformation of $\mathbf R ^ {n}$. For $n > 2$ and $k > 0$, this was proved by J. Lester; for $n = 2$ and for $n > 2$ and $k < 0$ this was proved by W. Benz (see [a2] for all these results). The proofs are different for the three cases, and no common proof is known (1996).

Let $X$ and $Y$ be normed real vector spaces such that $Y$ is strictly convex and the dimension of $X$ is at least $2$ (cf. also Vector space; Convex set). Let $k > 0$ be a fixed real number and $N > 1$ a fixed integer. Suppose that $f : X \rightarrow Y$ is a mapping satisfying

$$\left \| {a - b } \right \| = k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \leq k,$$

$$\left \| {a - b } \right \| = N k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \geq N k ,$$

for all $a,b \in X$. Then $f$ is an isometric operator, and hence an affine transformation (the Benz–Berens theorem, [a2]). If $X$ and $Y$ are, in addition, pre-Hilbert spaces (cf. Pre-Hilbert space), then $N$ may be replaced by any real number greater than $\sqrt 3$ (Radó–Andreascu–Valcán theorem).

There are generalizations for finite planes [a9], rational or constructible curves (B. Farrahi), non-Euclidean spaces [a11], and planes over fields [a10]. F. Radó [a8] has proved the following theorem. Let $V$ be a non-singular metric vector space of dimension $\geq 3$ over ${ \mathop{\rm GF} } ( p ^ {m} )$, $p \neq2$, $m \geq 3$, and let $k \neq0$ be a fixed element of ${ \mathop{\rm GF} } ( p ^ {m} )$. If $f$ is a bijection of $V$ preserving distance $k$, then $f$ is a semi-affine mapping if $n \not\equiv0, -1, - 2 ( { \mathop{\rm mod} } p )$.

Let $f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} }$, $n \geq 3$, be a mapping satisfying

$$\forall a,b,c \in \mathbf R ^ {n} : \Delta ( a,b,c ) = 1 \Rightarrow$$

$$\Rightarrow \Delta ( f ( a ) ,f ( b ) ,f ( c ) ) = 1,$$

where $\Delta ( a,b,c )$ denotes the area of the triangle with vertices $a,b,c$. Then $f$ is a Euclidean motion, i.e., $x \rightarrow f ( x ) - f ( 0 )$ is in $O ( n, \mathbf R )$ (Lester's theorem), [a3]. For $n = 2$ the equi-affine mappings are characterized similarly.

Let $\pi : {M ^ {n} } \rightarrow {M ^ {n} }$ be a mapping of the set $M ^ {n}$ of lines in $\mathbf R ^ {n}$, $n \geq 2$, into itself such that whenever $a,b,c \in M ^ {n}$ are the lines making up the sides of a triangle of area $1$, then $\pi ( a ) , \pi ( b ) , \pi ( c )$ are also the sides of a triangle of area $1$. Then $\pi$ is induced by a Euclidean motion of $\mathbf R ^ {n}$ for $n \geq 3$ and by an equi-affine mapping if $n = 2$ (the Wen-ling Huang theorem, [a3]).

Let $k > 0$ be a fixed real number and $X$ a normed real vector space of dimension $\geq 3$. Let $f : X \rightarrow X$ be a function satisfying

$$\forall x,y \in X: \left \| {x - y } \right \| = k \Rightarrow$$

$$\Rightarrow x - y,f ( x ) - f ( y ) \textrm{ linearly independent } .$$

Then there are elements $\lambda \in \mathbf R$ and $t \in X$ such that $f ( x ) = \lambda x + t$ for all $x \in X$. This theorem was proved by Benz [a2] and, anew, by D. Laugwitz [a6].

General references for this area are [a2], [a3] and [a7].

How to Cite This Entry:
Beckman-Quarles-type theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beckman-Quarles-type_theorems&oldid=52326
This article was adapted from an original article by W. Benz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article