Partitioning Euclidean Space

Transcription

1 Discrete Comput Geom 10: (1993) Springer-Verlag New York Inc. Partitioning Euclidean Space James H. Schmerl Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA Abstract. There is a partition of Euclidean n-space into countably many sets none of which contains the vertices of a regular n-simplex. A simple consequence of van der Waerden's theorem on arithmetic progressions is that, for any triangle T and any partition of the plane R 2 into finitely many pieces, there is a triangle similar to T whose vertices are in one of the pieces. This result, and its proof, easily extend to higher dimensions. For an integer n > 2 we consider Euclidean space ~" as a vector space over with the Euclidean norm Nxll and the usual inner product x.y. We call a subset T _ R" an n-simplex if I TI = n + 1 and T is not contained in any hyperplane. An n-simplex T is regular if there is some positive de R such that If x- Y I[ = d whenever x, y e T are distinct. Equivalently, T is regular iff (x - y). (x - z) = d2/2 whenever x, y, z e T are distinct. In particular, T ~ R 2 is a regular 2-simplex iff T is the set of vertices of an equilateral triangle, and T ~ 0~ 3 is a regular 3-simplex iff T is the set of vertices of a regular tetrahedron. Two n-simplices S, T are similar provided there is a function f: S--* T and some r > 0 such that [If(x) - f(y)[i = rl[x -- Y[I whenever x,y~s. The generalization of the aforementioned result can now be stated. If S _ ~" is an n-simplex, then, for any partition of R" into finitely many pieces, there is an n-simplex similar to S which is in one of the pieces. Complementing this result, Ceder [1] proved that R 2 can be partitioned into countably many pieces none of which contains all the vertices of an equilateral triangle. This was extended to ~3 by Komjfith [2] who proved, using a different technique, that R 3 can be partitioned into countably many pieces none of which contains the vertices of a regular tetrahedron. In this note we refine Komj~ith's technique to extend this result simultaneously in two ways: to arbitrary dimension

2 102 J.H. Schmerl n and to arbitrary n-simplices. We fix throughout an integer n > 2 which is to be the dimension of the space being considered. Theorem. Let S ~_ R" be any n-simplex. There is a partition of R" into countably many pieces none of which contains an n-simplex similar to S. The proof of the theorem uses the Axiom of Choice. Some use of the Axiom of Choice is necessary since, as Steinhaus first noted, for any n-simplex S _ ~", any X R" having positive Lebesgue measure contains an n-simplex similar to S. As a consequence we get that, for every n-simplex S _ R" and every partition of R" into countably many measurable sets, one of the sets contains an n-simplex similar to S. We identify the smallest infinite ordinal co with the set {0, 1, 2... } of natural numbers, and any natural number m we identify with the set {0, 1, 2... m - 1 } of smaller natural numbers. Let us fix some n-simplex S = {Xo, xt... x,} ~ R ". IfX _~ R ~ and if: X--* co, then we say that ~k isproper if there is no n-simplex T _ X which is similar to S such that I{~b(x): x e T}I = 1. The following is an obvious reformulation of the theorem. For every infinite cardinal x < 2~~ I RI and every real-closed field (.) F _ R such that IFt = x, there is a proper r F" --, co. To prove the theorem we prove (.) by induction on the cardinal x. For ~c = ~qo, the result is trivial: just let ~b: P' ~ co be any one-one function. To proceed by induction, assume that No < x < 2 ~~ and that whenever K ~_ is a real-closed field and IK[ < x, then there is a proper function 0: K ~ ~ co. Let F q R be real-closed such that tfi = x. We easily obtain a sequence (F~: 0t < x) such that the following four conditions hold: (1) S q F~; (2) if~t < fl <, then F~ and F B are real-closed fields, F= ~ Fa _ F and IFal < x; (3) if/~ < x is a limit ordinal, then Fp = U~ < ~ F=; (4) F = U= < ~ F=. We construct a sequence (~=: = < x) of proper functions ~k=: F=" -* co such that whenever ~ < p < x, then ~,= = ~,plf~". Clearly, then ~b = U, < ~ ~k~ will be a proper function r b TM ~ co. The sequence (~,=: = < x) is constructed by transfinite recursion. To that end, assume that 7 < x and that we already have (~,=: = < 7). We wish to construct ~k~: If 7 = 0, then let r F~ --* 09 be any proper function (the existence of which is guaranteed by the inductive hypothesis on ~). For 7 a limit ordinal, we have no choice but to set ~,r = ~,<~ ~b,. It remains to consider the case that 7 = ~ + 1. We need two functions to define ~. The first can be any proper function 0: F~ --+ co, the existence of which is guaranteed by the inductive hypothesis on ~r The second is a function tp as described in the next lemma.

3 Partitioning Euclidean Space 103 Lemma 1. There is m ~ co and a function cp: F~\F"- --* m with the following property: whenever T ~_ F~ is an n-simplex similar to S such that T c~ F", ~ (~ v ~ T\F~, then there are x, y ~ T\F~ such that r ~ tp(y). Before proving Lemma 1, we show how to obtain ~,y. For x ~ F~, let ~(x) if xe F], ~r(x) = (too(x) + tp(x) if x ~ F]. It should be clear that ~k~: F~ ~ co is proper and that ~k~ = ~brlf~-. Several more lemmas are needed to prove Lemma 1. Let S~-~= {x e R~: I[xll = 1} be the unit (n- 1)-sphere. The next lemma is an immediate consequence of the compactness of S n- ~. Lemma 2. For every e > 0 there is m ~ co and a function X: S"- 1 ~ m such that whenever x, y ~ S ~- 1 and Z(x) = XfY), then x " y >_ 1 - e. The next lemma is suggested by [2], and the proof presented here is also influenced by [2]. Recall that we have a fixed n-simplex S = {x o, xl... x,}. Lemma 3. There exists eo > 0 such that whenever T = {Yo, Yl... Yn} ~- Rn is an n-simplex similar to S for which xi = Yi for some i < n, xi ~ Yi for some i < n, Ilx~ - yi[i < eo for all i ~ n, then there are (necessarily distinct) i,j < n such that x~ ~ Yi, xj ~ yj, and yi--xi yj--xj I[Y, - x~ll [ly~- xjll < 1 -Co. Proof Intending to arrive at a contradiction, assume that there is no such Co. Then we can assume, without loss of generality, that Xo = 0 and for each integer m > 1 there is an n-simplex {Yore, Y1... Yn,} and rm > 0 such that, for all i,j<n, Yore ~ 0~ Ylrn =fi Xl, 1 IlYim - xi[i < -, m HYi. - Yj.II = r, llx~ - xjl[,

4 104 J.H. Schmerl and (Ylm--X~)'(Yjm--Xj)>--]lY~m-x~ll I[yjm-xjl](1 1). Moreover, without loss of generality we can assume that IlYlm - xx I[ ~ IlY~m - xilq for all i < n. For each m > 1, let d m = I[Ylm -- Xl II and Vim = (1/dm)/(Yim -- xi). Observe that Ilvlmll -< IlVlml[ = 1 and limmo dm= 0. Thus, without loss of generality, we can assume that the limits v~ = limm-.~ vim exist, and that v~ = bivl where 0 < b i < 1. Obviously, Ilvl II = 1. For i,j < n, let al = Ilxall and cij = xi" xj. Clearly, for m > 1 and i #j we have that and 2 2 rmai = Ilyimll 2 = IIx/ + dmvimll 2 : a 2 + 2dmxi'Vim + d211vlmll 2 (1) From (1) we get r2 cij = Yim " Yjm = (xi + dmvlm) " (xj + dmvjm ) 2 = Cij + drag i " Vjm + dmg j " Vim + dmvim Vjm. (2) r2,2~2 rn~i Ul = a2(a 2 + 2dmx, "Vim + d~mllvimll z) = a2(a2~ + 2dmx~ "V~m + d~llvxmli2), from which we get a 2 a 2 a2xi " V,m -- a2xx " Vlm = --~ dmllvxmlf z - ~ d.llvi.ii ~. Taking the limit as m ~ ~, we see that the right-hand side approaches 0, so we get a2xl " vl = a 2 xi " vi. (3) Similarly, from (1) and (2) we get from which we get 2 2 rmc,al = cil(a 2 + 2dmxl "vlm + d~llvxml[ 2) z = a~(c, + d.xi" vi. + dmxl" Vim + dmvim" vl~), 2c.(xi " v,~ - a2(x, "vi. + xl " V,m) = az dmv,m "Vim -- cil dm[ivlml] 2.

5 Partitioning Euclidean Space 105 Taking the limit as m ~ oo, we again see that the right-hand side approaches 0, so we get a2(xi " vl a t- x 1 9 vi) = 2Cil(X 1 "Vl). (4) Replacing vi by b~v~ in (3) and (4) yields respectively a2xl " vl = (a2xi) " (blvl) (5) and a2xi. v Cl,X 1 9 v I -- a2xl.biv 1. (6) Therefore, plugging (6) into the right-hand side of (5) yields or, equivalently, a2xl. v 1 = bi(2clix 1 9 v 1 - a2bixl. vl) (xl " vl)(a2b 2-2clibl + a 2) = O. We wish to conclude that xl 9 v~ = 0. If not, then b~ satisfies the equation a~x2--2cli x + a~ = O, so that 4c2i-4a~a 2 > O. However, since Xo, Xl,X i are not collinear, the 2 2 Cauchy-Schwarz inequality implies c2i < ala~, a contradiction. Thus, we have X 1 9 V 1 ~ 0. Now from (6) we get that x i 9 v 1 = 0 for all i. Since {x~, x2... x,} spans R n, this implies vl = 0, contradicting that tlvlll = 1. [] If X _ R" and x e R n, we say that x is algebraic over X if each coordinate of x is algebraic over the set of all coordinates of elements in X. Lemma 4. Let Y, Z ~_ R n be finite sets such that S ~_ Z and no x e Y is algebraic over Z. Let e > O. Then there is a function g: YuZ--*R ~ such that whenever T~_ Yu Z is an n-simplex similar to S, then so is g[t]; if xe Y, then 0 < [Ix -- g(x)[i < e; and ifx~z, then g(x) = x. Proof. This is just a special case of the following well known and easy consequence of Tarski's elimination of quantifiers for the first-order theory of R considered as an ordered field: If m e co, F ~ R is a real-closed field, X ~_ R m is first-order definable using parameters in F, (a o, al... am-1) ~ X, and e > 0, then there is (b o, b I... b,,_ ~ ) e X such that, for each i < d, [b i - all < e and if bi = a~, then ai ~ F. []

6 106 J.H. Schmerl We now finish the proof of the theorem by proving Lemma 1. Let m and Z be as in [,emma 2, using e = e o from Lemma 3. By compactness, it suffices to show that for each finite X F~ there is a function ~Oo: X~F] ~m with the following property: whenever T~_ X is an n-simplex similar to S such that T n F~ ~ ~ ~ T~F~, then there are x, y e T~F~ such that ~po(x) ~ ~Po(Y). Consider some finite X_ F~ such that S ~_ X, and let Y = X~F] and Z = Xn F~. Let r > 0 be small enough so that ify 1, Y2 ~ X are distinct and 0 < i < j < n, then, IIYl - Y2I[ > rl[xi - xjll and let ~ = r~o. Now apply Lemma 4 to get g: X~ ~ satisfying the properties specified in that lemma. For each x ~ Y, let (x-g(x) h ~po(x) = z Ilx g(x)ll] To see that r works, consider some n-simplex T _~ X which is similar to S such that T c~ Z # ~ ~ T n Y Clearly, by Lemma 3, there are x, y ~ T n Y such that x-g(x) y-g@) Ilx -- g(x)ll liy -- g(y)ll < 1 --%; hence, by Lemma 2, ~ao(x) # ~Po(Y). This completes the proof of Lemma 1, and, consequently, also of the theorem. References 1. Coder, J., Finite subsets and countable decompositions of Euclidean spaces, Rev. Roumaine Math. Pures Appl. 14 (1969), Komj/Ith, P., Tetrahedron free decomposition of R s, Bull. London Math. Soc. 23 (1991), ReceivedMarch 9, 1992.