A 2-metric characterization of the euclidean plane

Math. Ann. 206, 285--294 (1973) © by Springer-Verlag 1973

A 2-metric Characterization of the Euclidean Plane Raymond W. Freese 1. Introduction

It is known that the euclidean plane can be characterized metrically among the class of all metric spaces by each of several different sets of conditions. In particular, it is shown in [2] that a metric space is congruent with the euclidean plane if and only if it is complete, metrically convex, externally convex and has the property that each 4 of its points are congruently imbeddable in the euclidean plane, with some quadruple non-collinear. A similar question can be considered in a 2-metric (area metric) space ([4-6]) and the purpose of this paper is to prove that under suitable conditions of completeness, convexity and imbeddability of finite sets of points, a 2-metric space may be placed into a one-to-one, area preserving correspondence with the euclidean plane.

2. Preliminary Notions

By a 2-metric space is meant a set S of points a, b, c..... p, q, r .... and a function pqr, called the 2-metric or area, on ordered triples of points of S into the non-negative real numbers, satisfying (1) If p, qe S, there is a point r e S with pqra¢O, and (2) Each four points of S are 2-congruently imbeddable in the 3-dimensional Euclidean space E3, i.e., if p, q, r, s ~ S there are points p', q', r', s' e E3 and a 1- i area-preserving correspondence between the quadruples. A triple p, q, r of points of a 2-metric space is said to be linear provided pqr=O, A 1-1 area-preserving function between subsets of 2-metric spaces is called a 2-congruence, denoted by "'~2". The expression T ~ C 2 Ea indicates that the set T is 2-congruent with a subset of E3, and the notation Pl, P2 . . . . . Pn ~ 2Pl, P~..... P'n indicates that the n-tuples are 2-congruent in the given order. Several notions of betweenness can be defined in a 2-metric space. One of these is the notion of linear betweenness with respect to a point

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Bt(p, q, r) defined to mean p ~ q =t=r 4=p, prt 4: 0, pqr = 0 and tpq + tqr = tpr. Also used in the paper is the concept of linear betweenness B(p, q, r), defined to mean p ~ q ~ - r ~ p and for each t in S, t p q + t q r = t p r . A subset of a 2-metric space is said to be linearly 2-convex provided for each pair p, r of its distinct points, it contains a point q satisfying B(p, q, r) and is said to be linearly externally 2-convex provided for each pair p, q of its distinct points, there exists a point r of the subset such that B(/~, q, r). A weaker but more natural notion of betweenness is the notion of interior of a triple, defined as follows. A point p of a 2-metric space S is said to be weakly interior to q, r, s ~ S, denoted p'[qrs, provided pqr + p r s + p q s = qrs. If none of the areas involved vanishes we say that p is strictly interior to q, r, s and write pIqrs. A quadruple of distinct points is called triadic provided one of its points is weakly interior to the remaining triple, pairs of whose points are called the sides of the triadic quadruple. A subset of a 2-metric space is said to be externally 2-convex provided for each triple p, q, r of its points such that pqr~-O, there exists an s of the set such that rlpqs. A subset of a 2-metric space will be called a 2-segment (with vertices p, q, r) and denoted $2(p, q, r) provided it is 2-congruent with a closed euclidean triangle and will be called a 2-line, denoted by L2(p, q, r) provided it is 2-cong_ruent with the euclidean plane. Similarly, a 1-segment (with endpoints p, q), denoted S1 (P, q), is the set of all x in the space such that Bt(p, x, q) for all t such that pqt ~ O, while a l-line Ll(p, q) is the set of all x of the space such that pqx =0. Similarly we can define a number of notions of convergence of a sequence of points. A sequence {p~} of points of a 2-metric space is called weakly 2-convergent to p in S provided limp, pt = 0 for each point t of S. The sequence {Pn} is said to be a 2-Cauchy sequence provided for some non-linear triple a, b, c s S we have lim apmpn = lim bpr, p. = lim cp,~pn = 0

(m, n ~ oo).

A simple example shows that a weakly 2-convergent sequence need not be 2-Cauchy, and that the 2-metric is not necessarily a continuous function relative to the weak 2-convergence topology. The notion of strong 2-convergence is therefore introduced as follows. The sequence {p,} C S is said to be strongly 2-convergent to p ~ S provided (1) {Pn} is weakly 2-convergent to p, and (2) for each point q e S and each sequence {q~} weakly convergent to q, we have lim pp,~ q~= 0 (ra, n ~ oo). It is now noted that every strongly 2-convergent sequence is a 2-Cauchy sequence, and that relative to the strong 2-convergence topology, the 2-metric function is continuous. More surprising is the

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fact that in any 2-metric space with continuous 2-metric, strong 2-convergence and weak 2-convergence are equivalent. Clearly in euclidean spaces the notions of weak 2-convergence, strong 2-convergence, and metric convergence are equivalent, as are the notions of 2-Cauchy and metrically Cauchy sequences. Defining a 2-metric space S as 2-complete provided every 2-Cauchy sequence of its points is strongly 2-convergent to a point of the space, and defining a 2-segment as above, the main result may now be stated. Theorem. Let M be a 2-complete, linearly 2-convex, externally 2-convex 2-metric space in which each quadruple of its points is 2-congruent with a quadruple of E 2 and in which each 5 points containing a triadic quadruple and a point linear with one side are 2-congruent with 5 points of E 3. Then M ,-~2E 2. In the remainder of the paper, the phrase "a space M" shall mean a 2-complete, linearly 2-convex 2-metric space in which each 5 points containing a triadic quadruple and a point linear with one side are 2-congruent with 5 points of E 3. It is to be noted that this 5 point property is quite analogous to the weak Euclidean 4-point property for a metric space introduced by L.M.Blumenthal which states that each quadruple of its points which contains a linear triple is congruent with 4 points of the euclidean plane.

3. Properties of 1-Segments and 1-Lines Murphy has shown in [7] that (restated using the above notation)

Ll(a, b) becomes a metric space under d(x, y)= pxy for any p such that pab 4:0 and that this space is complete and convex if the 2-metric space is linearly 2-convex and 2-complete. Hence because of properties of metric betweenness and metric segments I2] as well as the imbeddability of each 4 points of M into Ea, the following properties are readily verified in a space M. Property I. Bp(a, b, c) and Bp(b, c, d) imply Bp(a, b, d) and Bp(a, c, d). Property 2. Bp(a, b, d) and Bp(b, c, d) imply Bp(a, b, c) and Bp(a, c, d). Property 3. Bp(a, b, c) holds if and only if Bp(c, b, a). Property 4. If abc= 0 and abp 4:0 but a 4: c 4: b, then one of the following hold: Bp(a, b, c), Bp(b, a, c), Bp(a, c, b). Property 5. If a, b, p are points of M such that abp 4: O, then {x ~ M I pax + p xb = pab, xab = 0} is a metric segment under the induced metric. Property 6. If Bp(a, b, c) holds, then Bp(b, a, c) does not hold.

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Using these properties, a sequence of lemmas will now be proven, developing further characteristics of a space M.

4. Properties of 2-Segraents and 2-Lines Lennna 1. I f p, q, r are points of a space M such that pqr 4: 0, there do not exist points s, s', s4: s', such that p, q, r, s.~2p, q, r, s' with qrs=O.

Proof. Since qrs = 0, if there existed such a pair of points s, s', then qrs' = 0 and by imbedding the quadruple q, r, s, s' in E a it follows that s'rs = qss' = 0. By Property 4, By( q, r, s) or By(r, q, s) or By(q, s, r) holds. Since in general Bw(x , y, z) implies ylwxz, then in each of these cases p, q, r, s, s' consists of a triadic quadruple with a point linear with a side, yielding points p*, q*, r*, s*, s" of E 3 such that p, q, r, s, s' ,~ 2P*, q*, r*, s*, s". But in each ease the linearity of q*, r*, s" and the relationship between the euclidean areas would require s" to be the point s ~. Thus 0 = p*s"s* = pss' which with qss' = qrs = qrs' = 0 yields pqr = O. Hence the points s, s' are not distinct. A useful concept to be utilized temporarily is that of a 1-segment with respect to a point. If pab 4: O, then 1-segment (with endpoints a, b) with respect to p, denoted Sf(a, b), is the set of all points x of M such that Bp(a, x, b), together with a, b. Lenuna 2. I f x, y, z are points of a space M such that Bp(x, z, y,), then for all q of M such that qxy > O, Bq(x, z, y) holds.

Proof. The proof is divided into two cases. Case I. There exists a w eS'~(p,q) such that xyw=O. Suppose x 4: w 4: y. Then since p x y ~ 0 with xyw = 0, either Bp(x, y, w) or Bp(y, x, w) or Bv(x, w, y). If By{x, y, w) it is sufficient to observe that yIpxw, q is linear with p, w and hence p, y, w, x, q,~2P', Y', w', x', q' CE 3. Hence Bp(x, y, w) implies Bv.(x', y', w'). In E s, Bv.(x', y', w') implies Bq.(x', y', w') which again by the 2-congruence implies Ba(x, y, w). The proof in the event By(y, x, w) holds is identical. In the final situation in which B~(x, w, y) holds, suppose the contrary of the desired conclusion, i.e., Bg(x, w, y) does not hold. Now pxw > 0 implies qxw > 0 and hence by Property 4, Ba(x, y, w) or Ba(y, x, w) hold. But these imply, by the preceding two subcases that Bp(x, y, w) or Bp(y, x, w) subsist, contrary to fact. A similar proof, under the assumption that x ~=w ~ z will yield the equivalence of Bp(x, w, z), Ba(x, w, z), of By(x, z, w), Ba(x, z, w) and of Bp(w, x, z), Ba(w, x, zj. Corresponding equivalences hold if we assume y 4=w 4: z. Now even if w is not distinct from x, y, z, appropriate use of properties I, 2, 3, 4 in various possibilities shown that B~(x, z, y) subsists.

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Case 2. For all w • S~(p, q), x y w 4: O. Let A = { r • S~(p, q)IB,(x, z, y)} and let A* denote S'~(p, q) - A . Note that since x y w ~-0, w • A* implies Bw(x, y, z ) o r Bw(y,x, z). Considering S~(p, q) as a metric space and letting K = l u b {kl d(p, r ) ~ k and r~S'~(p, q)=~r•A}, we have O
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Corollary 1. There exists exactly 1 2-se#ment S2(P, q, r) with a given set of 3 non-collinear vertices p, q, r. Corollary 2. slpqr implies s ~ S2(p, q, r). Corollary 3. slpqr implies s ~ S2(p, q, 1").

Corollary 4. sl pqr implies there exists a t such that B(q, t, r) and B(p, s, t). Corollary 1 follows immediately from the theorem for if there existed a second 2-segment, then there would exist interior points that are not in the first 2-segment contrary to the theorem. Corollary 2 is merely a restatement of the theorem in view of the existence of 2-segments. Corollary 3 follows from Corollary 2 by continuity of the 2-metric while Corollary 4 follows from the definition of I and the uniqueness of 2-segments. Lenuna 5. A space M is linearly externally 2-convex if and only if it is externally 2-convex.

Proof. If M is linearly externally 2-convex, then given p, q, r such that pqr ~ 0, let x ~ M such that By(q, r, s) and t such that B~(p, s, t). Then by Lemma 3, rlpqt. Hence M is externally 2-convex. Suppose M is externally 2-convex. Let q, r be two distinct elements of M and let p e M such that pqr 4=O. Further let r* denote a point such that rIpqr*. Then by Corollary 4 of Lemma 4 there exists t such that B(q, r, t). The existence of the point t verifies M is linearly externally 2-convex.

Lemma 6, In a space M, B(p, q, r) implies that for all x such that pqx:~O, S2(P, q, x) is contained in S2(p, r, x). Proof. Let y e S2(p, q, x). Then y l p q x which with Bx(p, q, r) implies (by the five point property) that p, q, x, y, r ~ C2 Ea. Therefore y l p x r which by Corollary 3 of Lemma 4 implies y e S2(p, x, r). Hence S2(p, q, x) is contained in S2(p, r, x).

Lexmna 7. In a space M, sipqr, pqs ~=0 implies S2(P, q, s) is contained in S2(p, q, r). Proof. If qrs=O or prs=O, s4~r, then the conclusion follows by Lemma 6. If slpqr, then by Corollary 4 of Lemma 4, there exists a t such that B(q, t, r) and B(q, s, t). Then by two applications of Lemma 6, the proof is complete. There is an immediate Corollary. In a space M, x'[pqr and y l p q x implies yIpqr.

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Lemma 8. In a space M that is externally 2-convex, 9iven h > 0, and p, q, r such that p q r > 0 , there exists s e M such that B(q, r, s) and prs = h " pqr. Proof. Using the result of M u r p h y that { x l q r x = 0} is a metric space under d(u, v ) = puv, it then follows from the fact that M is linearly externally convex (by L e m m a 5) that the metric space is externally convex. Hence, since it is metrically complete as well as metrically convex it contains a metric line [2]. Therefore given q, r of this line, there exists an s of the line such that d(q, s) = d(q, r) + d(r, s) = (l + h). d(q, r). In other words, qrs = 0 and pqs = p q r +prs, prs, p r s = h . p q r which was to be shown.

Lemma 9. In a space M, if s l u v w and B(s, u, t), then s l t v w . Proof. Since s l u v w , s e S2(u, v, w) by L e m m a 4, Corollary 2. Hence there exists a z of S2(u, v, w) such that B(u, s, z) and B(v, z, w). Therefore B(t, u, z) holds, which with B(v, z, w) implies by L e m m a 3 that uItwv.

Lemma I0. In a space M that is externally 2-convex, oiven x ~ S2(pa, ql, rt) such that 3plql x = 3pl r 1x = 3 q l r 1x = p x q l r l , then there exist P2, q2, r2 sNch that: (i) S2(Pl , ql, rt) C $2(P2, q2, r2), (ii) p2q2r2 = 4 p l q l r I , (iii) p2q2r2 = 3xr2q 2 = 3xqzp2 = 3xpEr 2. Proof. Let y e S2(Pl, ql, rl) such that B(p 1, y, q0, B(rl, x, y) hold. Then the five point property applied to Pl, ql, rl, x, y implies Pl r xY = q 1r l Y and P l x y = q lxy. Let r 2 be such that (Lemma 8) B(x, r 1, r2) and p ~r i r2 - Pl x r 1- Then the five point property applied to pl, r2, Y, r i, q 1 implies q l r 2 y = p l r 2 y = ( 5 / 3 ) p l r l y = ( 5 / 3 ) q l r l y . This, together with B(r 2, r 1, y) and q l r l y = ( 3 / 2 ) q l r l x gives ql xr2 = 2 q t r l r 2 = 2 q l r l x . Hence we have shown that there exists a point r2 such that qlr2x =plr2x = 2plrlr 2 = 2qlrlr 2 = 2qlrlx = 2plrtx. In an identical manner, there exists a P2 such that q l p 2 x = r l p 2 x =2qlplPE=2rlplp2=2q~plx=2rlptx and a q2 such that plq2 x = rlq2x = 2rlqlq 2 = 2plqlq2 = 2rlqlx = 2plqlx. Then since the five point property can be applied to p~, P2, ql, q2, x, it fotlows that p 2 q 2 x = 2 p 2 x q l = 2 p ~ x q 2 . Similarly we have that p2r2x - - 2 p 2 x r 1 = 2p~ x r 2 and q2r2x = 2q2xrx = 2qt xr 2. Hence x q 2 r 2 = x p 2 q 2 = x p 2 r 2 = 4 x p t q l . N o w x l p x q l r ~ and B(X, pl,P2 ) implies by L e m m a 9 that x l p 2 r ~ q 1. This with B(x, r 1, r2) implies as before x l p 2r 2q 2 . Finally, this with B(x, q l , q2)implies x l p 2q 2r 2. F r o m xlp2q2r2 it follows by L e m m a 4 that x e $2(P2, qz, r2), But from the 2-congruence of $2(P2 , q2, r2) with a closed euclidean triangle, this

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implies $2(P2, q2, r2) -----$2(P2, q2, x)L-JS2(P2, r2, x)~S2(q2, r2, x). Since by Lemma 7, $2(pl, x, q l ) Z S 2 ( P 2 , X, q2), S2(Pl, x, rl)CS2(P2 , x, r2) , S2(ql , x, rl) C $2(q2, x, r2) , then S2(P 1, ql, rl) = S2(Pt, x, ql)~S2(pl, x, rl) u S 2 ( q l , x, rt)CS2(P2 , q2, r2). Hence there exist the points P2, q2, r2 as required. Corollary 1. Given p, q, r e M least one 2-line (plane) containing This follows from Lemma 10 respondence with the points of the

such that pqr ~ O, then there exists at p, q, r. by observing that the one-to-one corEuclidean plane induced by

L2(p, q, r) = U S2(Pi, ql, r~) i=l_ is a 2-congruence.

Corollary2. Given p, q e L 2 ( a , b , c ) and x e M such that pqx=O. Then x e L2(a, b, c). I f x ~ L2(a, b, c), then, since p, q, r, x ~ C2 E2 with pqx = 0, then there exists an x' e L2(a, b, c) such that p, q, r, x' ~ p, q, r, x contrary to Lemma I. Lemma 11. I f p, q, r are elements of a space M, where M is externally 2-convex and pqr4:O, there exists exactly one 2-line containing p, q, r. Proof. Suppose the contrary, i.e., there exist L2(P, q, r) and L~(p, q, r) such that L2(p, q, r) oe L~(p, q, r). Let s' e L~(p, q, r) - L2(p, q, r) and consider Si(s', x) where xlpqr, i.e., x e S2(p, q, r) C L2(P, q, r)c~L'2(p, q, r). Let s be a point of L2(p, q, r) such that S2(P, q, r)u {s} ~2S2(P, q, r) u {s'}, using the 2-congruence of L~ with L 2 and let y e Sl(s, x)c~S2(p, q, r) (possible since xlpqr). Then let w e {p, q, r} such that wxs 4: O. Then by the preceding 2-congruence, x, y, s, w ~ 2x, y, s', w which with xys = 0 is contrary to Lemma 1. Since the goal of this paper is to characterize the euclidean plane in terms of the 2-metric among the class of 2-metric spaces, one should observe that the conditions imposed upon M to this point are inadequate since E3 satisfies all these conditions yet is clearly not 2-congruent with the plane. Hence another condition is required. Property P. A 2-metric space S is said to have property P provided every 4 points of S are 2-congruent with some 4 points of E2.

Theorem. Let M be a 2.complete, linearly 2-convex, externally 2-convex 2-metric space which possesses property P and in which each 5 points contatnino a triadic quadruple and a point linear with one side are 2-congruent with 5 points of E 3. Then M ~2E 2.

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Proof. Since the euclidean plane under the usual 2-metric satisfies the conditions it is sufficient to show that if M possesses these properties, M,~2E2. By Lemma 11, there exists M* C M such that M* ~zE2 where M* = L2( p, q, r) for some p, q, r such that pqr ~ O. It remains to be shown that M - M* = 0. Let x ~ M - M*, and z ~ $2~, q, r) such that pqz = qrz = p r z = ( 1 / 3 ) p q r . By continuity of the 2-metric, there exists a point y ~ S l ( x , z ), x#:y=~z, such that pry<(1/2)pqr, pqy<(1/2)pqr, qry < (1/2) pqr. By property P, p, q, r, y ~ C2 E2 but at least one of the above inequalities is violated unless ylpqr. Hence by Lemma 7, y ~ S2(P, q, r). Now y,z are both in L2( p, q, r) which, with x y z = 0 yields by Corollary 2 of Lemma 10 that x ~ Lz(p, q, r ) = M*, a contradiction. In view of the strength of property P it is of interest to note that it does not imply the five point property, even in the environment of "a space M", and hence that the five point property is necessary to prove the theorem. This is shown by the following example. Let the point set of S be the point set of the euclidean plane and let unprimed letters denote points of S and the primed letters the identical points of E 2. Let T denote a fixed closed euclidean triangle with vertices a', b', c' and define x y z = x' y' z' + area of (T c~Sz(x', y', z')). Then dearly S is "a space M" and is externally 2-convex. Further, since every quadruple p, q, r, s of S is such that for some labeling pqr = pqs + qrs + prs or pqr + pqs = prs + qrs, then, since given any non-negative real numbers kl, kz, k3, k4, such that kl + k 2 = k 3 + k 4 or k 1 = k z + k3 + k4, there exists a quadruple of points of E2 with k 1, k 2, k3, k4 as its set of corresponding areas, every quadruple of S is 2-congruently imbeddable in E2. However, if d', e' are points such that c' is the midpoint of b', e' and d' is the midpoint of a', b', then if the five point property held, then a, b, c, d, e ~ 2a*, b*, c*, d*, e* of E2. Then since a, b, c, d ~, 2a*, b*, c*, d*, d* must be the midpoint of a*, b*. However, since a, b, c, e,~za*, b*, c*, e*, then d* must divide the segment from a* to b* in a 5 to 4 ratio, which is a contradiction.

References t. Andalafte,E.Z., Freese,R.W.: Existence of 2-segments in 2-metric spaces. Fund. Math., LX 201--208 (1967) 2. Blumenthal,L. M.: Theory and applications of distance geometry.Oxford: Clarendon Press, 1953 3. Cassens,P., Cassens,B. A., Frcese,R. W.: 2-metricaxiomsfor planeeuclideangeometry, Math. Nachr. 51, 11--24 (1971) 4. Freese,R. W,, Andalafte,E. Z, : A characterizationof 2-betweennessin 2-metric spaces. Canad. J. Math. 18. 963--968 (1966)

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5. G/ihler, S.: 2 metrische R~iume und ihre topologische Struktur. Math. Nachr. 26, 115--148 (1963----64) 6. Menger, K. : Untersuehungen fiber allgemeine Metrik. Math. Ann. 100, 75--163 (t928) 7. Murphy, G.P.: Lines in a planar space. Proc. Am. Math. Soc. 19, 1106---1108 (1968) Dr. Raymond Freese Department of Mathematics Saint Louis University St, Louis, Missouri 63103, USA

(Received September 9, 1972)