DEMONSTRATIO MATHEMATICA

Vol Xin No 4 1910

Jerzy Osetek

ON THE ORDER OF n+3 HYPERPLANES IN n-DIMENSIONAL PROJECTIVE GEOMETRY

In this paper we consider certain relations concerning linear varieties in an n-dimensional projective space. Let

Fn be the lattice of linear varieties in the n-dimensional projective space over the field of real numbers. By 0 and 1 we denote the minimal element of the lattice, i.e. the void variety, and the maximal element, i.e. the whole n-di- mensional space, respectively. We d-¡note by M^ the set of all k-dimensional (k = 0,1,2,... ,n-1) linear varieties, The elements of the sets M0, M-j and are called points, lines and hyperplanes, respectively. The lattice operations are denoted as usual by U and 0 .

For s >n and p^ , p2,... , pg e M0> we say that these points are in general position, iff there is no hyperplane h e Mn_i containing n+1 points from among p^,p2,..•,pg. Then p • U p. U ... U p.- =1 for each permutation 11 2 n+1

(1) c i^ ,i2,... ,i0> of s-tuple < 1,2,... ,s> .

Dually we introduce the general position of s-hyperplanes h^ ,h2,... ,hg e Mn_i (a >n), which holds whenever

h, n h. n ... nh. = c 11 2 n+1 for each permutation (1) of indexes.

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In [l] definitions of the relation ordering into cycles each n+3 p-ints in general position are given and the dual relation for hyperplanes is defined. Apart from the case n=1 and n=2 no geometrical intuition connected with these notions so far has been given. I'-'-.ote by D the relation defined in [l] for n+3 hyper- planes in n-dimensional space. I'he theorems Tn, 2n and 4n from [2] imply, in particular, that any n+3 hyperplanes

(2) h1,h2,...thn+3e Mn-1 satisfying the additional condition that

(3) h1,h2,...,hn+2 are in general position, are^orderable exactly in one way (up to the cyclic change of indexes and to inversion of them) namely

11 x2 n+3 so that the following condition is satisfied

(h h h (4) Dnn i ' xi »•••» i ) 2 n+3 *

In this paper we show what geometrical sense is related to the relation :Dn, for an arbitrary n. We show that be- tween connected regions, arising from division of the projec- tive space by hyperplanes h^,h2,...,hn+^, there are exactly n+3 simplexes which are placed in such a way that the simple- xes touching each other by vertices contitute a cycle, and their order in the cycle reflexes the order of hyperplanes satisfying the condition (4). In order to present this result more precisely, we procede as follows.

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First of all, we fix in pQ the set of n+3 hyperplanes being in general position

(5) H = jh^»• • •

By £> we denote the set of all connected open regions, ob- tained from sections of the projective space by the hyperpla- nes from H. Each of the regions belonging to S is bounded by n+1, n+2, or n+3 hyperplanes belonging to H. A region s belonging to S is said to be a simplex when it is bounded exactly by n+1 hyperplanes from the set H, or to be more precise, v/hen there exists an (n+1)-tuple

in+3> such that

(7) (n+3)-tuple < ±1 ,ig,... ,in+3> is a permutation of the sequence < 1 ,2,... ,n+3> ajnd such that each point belonging to the boundary of s lies at least on one of the hyperplanes of the set. H', where

(8) H =j h. ,h. ,...,h. I x2 ^+1

In other words, s e S is a simplex when there exists a se- quence (6) satisfying the condition (7), such that none of two hyperplanes

(9) h. ,h. xn+2 n+3 has common points with the closure of the region s. In this case we say that hyperplanes (8) "determine the simplex s" or alternatively, that they are faces of this simplex, we deno- te the simplex by

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(10) s(i1ti2,...,in+1).

Of course, the order of arguments in (10) is not important. In case n=2, the set S contains always eleven regions, fi- ve among which are simplexes. Such an example is presented in fig,1e The simplexes are distinguished in the picture by sha- dowing and are denoted by s1 ,sp,... ,St-.

Prom condition (3) it follows that any n different hy- per planes

(11) h± ,h± ,...,h± 12 n from the set H meet in one point

(12) p = h. n h. n ... n h. . 11 2 xn we say that the point p is "determined by hyperplanes" (11). In particular, if hyperplanes (11) are faces of a certain sim- plex s e S then p is said to be a vertex of the simplex s. It is evident that every simplex has exactly n+1 vertices. Hence, the vertices of the simplex will be the points

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(13) p, = h. n h. n „.nh. ol n ...nh, ; 3 X1 2 lj-1 j+1 n+1 j=1,2,...,n+1.

Similarly from the condition (3) it follows that any n-1 dif- ferent hyperplanes

(14) h, ,h, ....»h, X1 x2 n-1 from H meet one another along one line

i = h. n h, n ... n h. X1 2 ln-1

This line ia said to be "determined by hyperplanes" (14). Now, we shall prove the following lemma. Lemma 1. If (7) holds then hyperplanes (8) deter- mine a simplex belonging to £> iff the condition

(15) c(p, ,p. ; (p. U p, )n h. , (p, U Pi ¡nh. ) \ 31 d2 •s^ J2 n+2 J1 a2 n+3/ holds for each pair p. , p. of points determined by hyper- 31 32 planes belonging to H', (C denotes separation of pairs of points on the line). Proof. The hyperplanes belonging to the set h' decompose the projective space into k=2n connected regions s1 ,s2,... ,sk. I'hese hyperplanes determine precisely n+1 points

(16) P-) » P2 » • • • • PQ+1 defined by formula (13). Moreover, they determine m = lines

(17) l1tl2 lm.

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On each of these lines 1. there are two points p. , p. from among the points (16), These points ¿ivide the line li into two projective segments, ¿ach of the regions s. has J m-edges which are just such projective segments - one from each of lines li. If condition (15) did not hold at least for one of L = pi U p, , then each of two projective seg- i j-j J2 ments on the line (with ends p. , p. ) would be cut by one of 3-1 J2 the hyperplanes h. ,h., and therefore none of the seg- n+2 ~n+3 ments could not be an edge of a simplex belonging to S. On the other hand, if condition (15) is satisfied by each pair of points determined by hyperplanes from the set H' then, as it will be proved, one of the regions s^ is divided nei- ther by h. nor by h. , hence it belongs to S. In or- n+2 n+3 der to prove this, it suffices to notice that adopting hH xn+2 as an improper hyperplane we obtain a division of the result- ing n-dimensional affine space by hyperplanes (8) into regions one of which is a simplex bounded by the hyperplanes. From con- dition (15) it follows that hyperplane h.Z n+3 does not meet the simplex, because it meets none of its edges. The main.result of our paper is expressed in the follow- ing theorem. Theorem. If hyperplanes (2) are in general posi- tion then exactly n+3 simplexes belong to the set S. If, furthermore, the condition

(20) Dn(h1,h2,...,hn+3) holds, then the simplexes belonging to S are the following ones: s1 = s(1,2,...,n+1), s2 = s(2,3,...,n+1,n+2),...,

= s(n+2,n+3,1 ,2,... ,n-1 ), sn4.-5 = s(n+3,1,2,...,n).

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In the case n=2 the following figure (fig.2) illustrates the the ore®.

In the proof of the theorem we use Lemma 1, Theorem 4n from [2] p.232, and Lemma 2, given below, which is a corrolla ry from Theorem 5 proved in [2] p.235. Lemma 2. If (20) holds and

(21 J i< j

(22) 1 = i13km = ii-tn ...n hi_1 n hi+1 n ... nk^n hj+1 n ...n

n n 0 0 n 0 0 \+1 • • •• V1 • •' • V3 then

d1 (in ^,10 h^in hk,in hm).

We give now the proof of our theorem. Prom Theorem 4n it follows that for the hyperplanes (2) there exists a permutation for which the relation Dn holds.

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After appropriate remuneration we can assume that condition (20) holds. Prom Lemma 2 we infer that for 1 = 1. . „,_ where 10,n+2,n+3' the following condition holds

Vinhi»inVinV2'inV3) and hence

c(inhifinhjj inhn+2,inhn+3).

Prom Lemma 1 it follows that hyperplanes h^ »hg,... de- termine a simplex s^ = a(1,2,...,n+1) belonging to S. The existence of the remaining simplexes in S, described in the conclusion of the theorem, can be proved in the some way. Por completeness of the proof it suffices to verify that apart from those simplexes, there exists no other one belonging to S. Assume that for a certain permutation (6) satisfying condition (7), the hyperplanes h. ,h. ,...,h. determine a simplex X1 x2 n+1 sQ belonging to S and different from the simplex s.|,s2,... ,s anc i cari ••* n+3* Hence in+2 * n+3 neither be consecutive natural numbers, nor numbers 1 and n+3. Therefore, without limiting generality of our considera- tions, we can assume that there exist j,m such that

1 * in+2* J < *n+3< n+3.

If we denote 1 = lj . „ then by Lemma 2 we obtain W'^n-O'"1

D^inL ,ini.,ini. ,ini ). 1 n+2 3 n+3

Therefore (according to the definition of D^ in [2] p.213) we havo

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(23) C(in h.,inhm;in h. ,1D h. ). J m n+2 n+3

Simultaneously, applying Lemma 1 to the simplex sQ we obtain the negation of condition (23). This ends the proof of the theorem.

REFERENCES

[I]L. Dubikajtis: On the order of points and hyperplanes in n-dimensional projective geometry, Bull. Acad. Polon. Sci., Se'r. sci. math., astr., phys. 6 (1958) 607-610. 2 L. Dubikajtis: Une extension de la notion d'ordre lineiare à celle d'ordre de dimension n, Ann.Polon. Math. 14 (1964) 211-238.

INSTITUTE OP MATHEMATICS, NICHOLAS COPERNICUS UNIVERSITY, T0RU1Î Received February 22, 1979.

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