|±||β|

If ± ∈ Rn , then

±⊥ = { β ∈ Rn | (±, β) = 0 }

in the linear subspace normal to ±. If ± = 0 then dim ±⊥ = n ’ 1.

A¬ne Euclidean space ARn

1.1.3

The real a¬ne Euclidean space ARn is simply the set of all n-tuples

a1 , . . . , an of real numbers; we call them points. If a = (a1 , . . . , an ) and

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 3

b = (b1 , . . . , bn ) are two points, the distance r(a, b) between them is de¬ned

by the formula

(a1 ’ b1 )2 + · · · + (an ’ bn )2 .

r(a, b) =

On of the most basic and standard facts in Mathematics states that this

distance satis¬es the usual axioms for a metric: for all a, b, c ∈ Rn ,

• r(a, b) 0;

• r(a, b) = 0 if and only if a = b;

• r(a, b) + r(b, c) r(a, c) (the Triangle Inequality).

With any two points a and b we can associate a vector2 in Rn

«

b1 ’ a 1

¬ ·

.

.

ab = .

.

bn ’ an

If a is a point and ± a vector, a + ± denotes the unique point b such that

ab = ±. The point a will be called the initial, b the terminal point of the

vector ab. Notice that

r(a, b) = |ab|.

The real Euclidean space Rn models what physicists call the system

of free vectors, i.e. physical quantities characterised by their magnitude

and direction, but whose application point is of no consequence. The n-

dimensional a¬ne Euclidean space ARn is a mathematical model of the

system of bound vectors, that is, vectors having ¬xed points of application.

1.1.4 A¬ne subspaces

Subspaces. If U is a vector subspace in Rn and a is a point in ARn then

the set

a + U = {a + β | β ∈ U }

is called an a¬ne subspace in ARn . The dimension dim A of the a¬ne sub-

space A = a + U is the dimension of the vector space U . The codimension

of an a¬ne subspace A is n ’ dim A.

2

It looks a bit awkward that we arrange the coordinates of points in rows, and the

coordinates of vectors in columns. The row notation is more convenient typographically,

but, since we use left notation for group actions, we have to use column vectors: if A is

a square matrix and ± a vector, the notation A± for the product of A and ± requires ±

to be a column vector.

4

If A is an a¬ne subspace and a ∈ A a point then the set of vectors

A = { ab | b ∈ A }

is a vector subspace in Rn ; it coincides with the set

{ bc | b, c ∈ A }

and thus does not depend on choice of the point a ∈ A. We shall call A

the vector space of A. Notice that A = a + A for any point a ∈ A. Two

a¬ne subspaces A and B of the same dimension are parallel if A = B.

Systems of linear equations. The standard theory of systems of si-

multaneous linear equaitions characterises a¬ne subspaces as solution sets

of systems of linear equations

a11 x1 + · · · + a1n xn = c1

a21 x1 + · · · + a2n xn = c2

. .

. .

. .

am1 x1 + · · · + amn xn = cm .

In particular, the intersection of a¬ne subspaces is either an a¬ne subspace

or the empty set. The codimension of the subspace given by the system of

linear equations is the maximal number of linearly independent equations

in the system.

Points. Points in ARn are 0-dimensional a¬ne subspaces.

Lines. A¬ne subspaces of dimension 1 are called straight lines or lines.

They have the form

a + R± = { a + t± | t ∈ R },

where a is a point and ± a non-zero vector. For any two distinct points

a, b ∈ ARn there is a unique line passing through them, that is, a + Rab.

The segment [a, b] is the set

[a, b] = { a + tab | 0 t 1 },

the interval (a, b) is the set

(a, b) = { a + tab | 0 < t < 1 }.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 5

Planes. Two dimensional a¬ne subspaces are called planes. If three

points a, b, c are not collinear , i.e. do not belong to a line, then there is a

unique plane containing them, namely, the plane

a + Rab + Rac = { a + uab + v ac | u, v, ∈ R }.

A plane contains, for any its two distinct points, the entire line connecting

them.

Hyperplanes, that is, a¬ne subspaces of codimension 1, are given by

equations

a1 x1 + · · · + an xn = c. (1.1)

If we represent the hyperplane in the vector form b + U , where U is a

(n ’ 1)-dimensional vector subspace of Rn , then U = ±⊥ , where

«

a1

¬.·

± = . .

.

an

Two hyperplanes are either parallel or intersect along an a¬ne subspace

of dimension n ’ 2.

1.1.5 Half spaces

If H is a hyperplane given by Equation 1.1 and we denote by f (x) the

linear function

f (x) = a1 x1 + · · · + an xn ’ c,

where x = (x1 , . . . , xn ), then the hyperplane divides the a¬ne space ARn

in two open half spaces V + and V ’ de¬ned by the inequalities f (x) > 0

+ ’

and f (x) < 0. The sets V and V de¬ned by the inequalities f (x) 0

and f (x) 0 are called closed half spaces. The half spaces are convex in

the following sense: if two points a and b belong to one half space, say, V +

then the restriction of f onto the segment

[a, b] = { a + tab | 0 t 1}

is a linear function of t which cannot take the value 0 on the segment

0 t 1. Hence, with any its two points a and b, a half space contains

the segment [a, b]. Subsets in ARn with this property are called convex.

More generally, a curve is an image of the segment [0, 1] of the real line

R under a continuous map from [0, 1] to ARn . In particular, a segment

[a, b] is a curve, the map being t ’ a + tab.

6

Two points a and b of a subset X ⊆ ARn are connected in X if there is

a curve in X containing both a and b. This is an equivalence relation, and

its classes are called connected components of X. A subset X is connected

if it consists of just one connected component, that is, any two points in

X can be connected by a curve belonging to X. Notice that any convex

set is connected; in particular, half spaces are connected.

If H is a hyperplane in ARn then its two open halfspaces V ’ and V +

are connected components of ARn H. Indeed, the halfspaces V + and V ’

are connected. But if we take two points a ∈ V + and b ∈ V ’ and consider

a curve

{ x(t) | t ∈ [0, 1] } ‚ ARn

connecting a = x(0) and b = x(1), then the continuous function f (x(t))

takes the values of opposite sign at the ends of the segment [0, 1] and thus

should take the value 0 at some point t0 , 0 < t0 < 1. But then the point

x(t0 ) of the curve belongs to the hyperplane H.

1.1.6 Bases and coordinates

Let A be an a¬ne subspace in ARn and dim A = k. If o ∈ A is an arbitrary

point and ±1 , . . . , ±k is an orthonormal basis in A then we can assign to

any point a ∈ A the coordinates (a1 , . . . , ak ) de¬ned by the rule

ai = (oa, ±i ), i = 1, . . . , k.

This turns A into an a¬ne Euclidean space of dimension k which can be

identi¬ed with ARk . Therefore everything that we said about ARn can be

applied to any a¬ne subspace of ARn .

We shall use change of coordinates in the proof of the following simple

fact.

Proposition 1.1.1 Let a and b be two distinct points in ARn . The set of

all points x equidistant from a and b, i.e. such that r(a, x) = r(b, x) is a

hyperplane normal to the segment [a, b] and passing through its midpoint.

Proof. Take the midpoint o of the segment [a, b] for the origin of an

orthonormal coordinate system in ARn , then the points a and b are rep-

resented by the vectors oa = ± and ob = ’±. If x is a point with

r(a, x) = r(b, x) then we have, for the vector χ = ox,

|χ ’ ±| = |χ + ±|,

(χ ’ ±, χ ’ ±) = (χ + ±, χ + ±),

(χ, χ) ’ 2(χ, ±) + (±, ±) = (χ, χ) + 2(χ, ±) + (±, ±),

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 7

which gives us

(χ, ±) = 0.

But this is the equation of the hyperplane normal to the vector ± directed

along the segment [a, b]. Obviously the hyperplane contains the midpoint

o of the segment.

1.1.7 Convex sets

Recall that a subset X ⊆ ARn is convex if it contains, with any points

x, y ∈ X, the segment [x, y] (Figure 1.7).

¨r

¨¨ rr

¨ r

¨ r xr

xr

d

d d

d d

dry

yr d

r ¨

r ¨ d

d

rr ¨¨

r¨

convex set non-convex set

Figure 1.1: Convex and non-convex sets.

Obviously the intersection of a collection of convex sets is convex. Every

convex set is connected. A¬ne subspaces (in particular, hyperplanes) and

half spaces in ARn are convex. If a set X is convex then so are its closure

X and interior X —¦ . If Y ⊆ ARn is a subset, it convex hull is de¬ned as

the intersection of all convex sets containing it; it is the smallest convex

set containing Y .

Exercises

1.1.1 Prove that the complement to a 1-dimensional linear subspace in the

2-dimensional complex vector space C2 is connected.

1.1.2 In a well known textbook on Geometry [Ber] the a¬ne Euclidean spaces

are de¬ned as triples (A, A, ¦), where A is an Euclidean vector space, A a set

and ¦ a faithful simply transitive action of the additive group of A on A [Ber,

vol. 1, pp. 55 and 241]. Try to understand why this is the same object as the

one we discussed in this section.

8

1.2 Hyperplane arrangements

This section follows the classical treatment of the subject by Bourbaki

[Bou], with slight changes in terminology. All the results mentioned in

this section are intuitively self-evident, at least after drawing a few simple

pictures. We omit some of the proofs which can be found in [Bou, Chap. V,

§1].

1.2.1 Chambers of a hyperplane arrangement

A ¬nite set Σ of hyperplanes in ARn is called a hyperplane arrangement.

We shall call hyperplanes in Σ walls of Σ.

Given an arrangement Σ, the hyperlanes in Σ cut the space ARn and

each other in pieces called faces, see the explicit de¬nition below. We wish

to develop a terminology for the description of relative position of faces

with respect to each other.

If H is a hyperplane in ARn , we say that two points a and b of ARn

are on the same side of H if both of them belong to one and the same of

two halfspaces V + , V ’ determined by H; a and b are similarly positioned

with respect to H if both of them belong simultaneously to either V + , H