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