finsler norms on a vector space

0

D.Kertész R.L.Lovas J.Szilasi

FINSLER NORMS ON A VECTOR SPACE a pedagogical introduction MOTTO 1 the hardest task in the world begins easy the greatest goal in the world begins small LAO-TZU

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MOTIVATION Finslerian version of Hilbert's fourth problem:

Given a FLAT SPRAY over a manifold. Find the Finsler functions whose canonical spray is PROJECTIVELY RELATED to the given spray. (J.Sz., DGA 2007, 539-558) M. Crampin, Some remarks on the Finslerian version of Hilbert's fourth problem (version July 22, 2008; appeared in Houston J. Math.)

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MOTTO 2 Herbert Busemann: `... the study of Minkowskian geometry ought to be the rst and main step, the passage from there to general Finsler spaces will be the second and simpler step` (The geometry of Finsler spaces, Bull. Amer. Math. Soc. 56 (1950), 5-16)

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MINKOWSKI SPACES

V a real vector space F : V → R, v 7−→ F (v)

a function

(N1)

F (u + v) ≤ F (u) + F (v)

subadditivity

(N2)

F (λu) = λF (u); λ ∈ R∗+, u ∈ V

positive homogeneity

(N2) ⇒ F (0) = 0 (N1) and (N2) F is sublinear

(N3)

symmetry (N2) and (N3) ⇔ F is absolutely homogeneous:

F (v) = F (−v)

F (λv) = |λ|F (v);

λ ∈ R, v ∈ V

 (N 1)  

(N 2)  (N 3) 

(N4)

F is a seminorm

⇒ F (v) ≥ 0

for all

F (v) = 0 ⇒ v = 0 deniteness (N1)-(N4) F is a norm on V (V, F ) satisfying (N1)-(N4) a normed

v∈V

space Minkowski space: a nite dimensional normed space

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BASIC ASSUMPTIONS VECTOR SPACE

=

nite dimensional (non-trivial) real vector space

Ã vector spaces have a 'natural' Hausdor topology

which makes them a topological vector space; this topology is unique (Tychono's theorem)

VECTOR SPACES ARE EQUIPPED WITH THEIR CANONICAL TOPOLOGY ∼◦∼

MOTTO 3 J. Dieudonné on linear algebra: `... generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.`

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DERIVATIVES IN VECTOR SPACES V, W vector spaces U ⊂ V nonempty open subset f : U −→ W is dierentiable at a mapping f 0(p) ∈ L(V, W ) such that

p ∈ U

f (p + tv) − f (p) = f 0(p)(v), t→0 t at every point of U , lim

If this holds

point

if there exists a linear

for all

v ∈ V.

f 0 : U −→ L(V, W ), p 7−→ f 0(p)

is the derivative of f . If f 00

f0

³

is dierentiable over U , its second derivative ´

∼ L2(V, W ), p 7−→ f 00(p) : U −→ L V, L(V, W ) =

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DERIVATIVES IN VECTOR SPACES General hypothesis as above. Schwarz's theorem If f is 2-times dierentiable at p, then f 00(p) ∈ L2 sym (V, W ), i.e., f 00(p)(u, v) = f 00(p)(v, u),

Taylor's theorem If p ∈ U,

and

v∈V

f

is

(k + 1)-times

is such that

for all

u, v ∈ V.

continuously dierentiable,

p + [0, 1]v ⊂ U ,

then

1 1 f (p + v) = f (p) + f 0(p)(v) + f 00(p)(v, v) + · · · + f (k)(p)(v, · · · , v)+ 2 k! +

1 f (k+1)(p + τ v)(v, . . . , v); τ ∈ ]0, 1[ (k + 1)!

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HOMOGENEOUS FUNCTIONS V a vector space U ⊂ V a nonempty open subset r∈R f : U −→ is r+-homogeneous if for

each

and

t ∈ R∗+

u∈U

we have

tu ∈ U, f (tu) = tr f (u).

FACTS Under the above general hypothesis: (1) if f is dierentiable at each point of U , then it is and only if f 0(v)(v) = rf (v)

for all

r+-homogeneous

if

v∈U

(Euler's relation); (2) if

f is r+-homogeneous, where r ∈ N, and r-times dierentiable 0 ∈ V , then f is a homogeneous polynomial of degree r.

at

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PRE-FINSLER NORMS Denition A function F

:V →R

norm if it is

on a real vector space V is a pre-Finsler

(i) non-zero; (ii)

1+-homogeneous;

(iii) of class

C2

on

V \ {0}.

Claim 1: If

F :V →R dim

If

λ∈R

and

³

is a pre-Finsler norm and

Ker(F 0(p))

λ p, v := F (p)

= n − 1, n = dim V.

then

F 0(p)(v) = ⇒ F 0(p) : V → R

´

F (p) 6= 0,

λ Euler λ F 0(p)(p) = F (p) = λ. F (p) F (p)

is surjective.

then

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PRE-FINSLER NORMS Claim 2: Hypothesis as above. F 00(p) ∈ L2 sym (V ), p ∈ V \ {0}

is of rank at most Euler ⇒ ⇒

n − 1.

F 0 : V \ {0} → V ∗is 0+-homogeneous

for all is degenerate.

F 00(p)(p, v) = 0 F 00(p)

Remark rank(F 00(p)) := rank(jF 00(p)), jF 00(p) : V −→ V ∗, u 7−→ jF 00(p)(u) jF 00(p)(u)(v) := F 00(p)(u, v)

p ∈ V \ {0}, v ∈ V

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OBJECTS ASSOCIATED TO A PRE-FINSLER NORM F : V → R

I Energy function

E :=

Properties (1)

E

is of class

(2)

E

is

(3)

E(v) ≥ 0

C2

on

V \ {0}, C 1

1 2 F 2

on

V;

2+-homogeneous;

for all

v ∈V,

and

E(v) = 0 ⇔ F (v) = 0.

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II Scalar product tensor eld

g : p ∈ V \ {0} 7−→ gp := E 00(p) ∈ L2 sym (V )

Properties (1)

gp(p, v) = E 0(p)(v) = F (p)F 0(p)(v); p ∈ V \ {0}, v ∈ E;

(2)

gp(p, p) = 2E(p) = F 2(p), p ∈ V \ {0};

(3)

gλp = gp; p ∈ V \ {0}, λ ∈ R∗+.

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III


F -scalar product h , iF : V \ {0} × V −→ R, (u, v) 7−→ hu, viF := gu(u, v)

def. u⊥F v ⇔ hu, viF = 0

Properties (1) h , iF is R-linear in its second variable, its rst variable; (2) hu, viF 6= hv, uiF , hence u⊥F v 6⇒ v⊥F u; (3) if F (u) 6= 0, then u⊥F v

if and only if

v ∈ Ker

1+-homogeneous

³

F 0(u)

´

;

(4) if def. u⊥B v ⇔ F (u) ≤ F (u + tv)

for all

t ∈ R.

(Birkho-orthogonality ), then u⊥B v ⇒ u⊥F v

(u ∈ V \ {0}, v ∈ V )

in

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IV Projection ¯tensor eld o n ¯

U := p ∈ V ¯F (p) 6= 0 ∼ P : U −→ T1 1 (V ) = End(V ), p 7−→ Pp ,

Pp(v) := v −

gp (p,v) p gp (p,p)

=v−

F 0(p)(v) p F (p)

Projection onto

³

span(p)

´⊥ g

def. (v⊥gp w ⇔ gp(v, w) = 0)

p

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OBJECTS ASSOCIATED TO A PRE-FINSLER NORM F : V → R Properties (1)

Ker(Pp) = span(p);

(2)

Pp2 = Pp,

(3) (4)

i.e., ³

Pp

is a projection operator in ´⊥g

³

p

F 0(p)

Im(Pp) = span(p) = Ker hence p⊥F Pp(v) for all v ∈ V ; V = span(p) ⊕ Ker

³

F 0(p)

´

.

´

,

V;

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V 'Angular metric' tensor eld ¯ n o ¯

U := p ∈ V ¯F (p) 6= 0

0 0 h : U −→ T0 2 (V ), p 7−→ hp := gp − F (p) ⊗ F (p)

Properties (1)

hp = F (p)F 00(p);

(2)

gp Pp(v), w = hp(v, w); v, w ∈ V (P and h are 'g-equivalent', can be obtained from P 'by lowering of an index');

(3)

³

³

´

´

gp Pp(v), Pp(w) = hp(v, w); v, w ∈ V .

Corollary If

F (p) 6= 0,

then

F 00(p)(v, w)

=

F 00(p)

³

´

Pp(v), Pp(w) ; v, w ∈ V.

or:

h

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C-REGULARITY Denition A pre-Finsler norm (i) C-regular at a point (

F :V →R

p ∈ V \ {0}

is

if

F 00(p)(v, v) ≥ 0 for all v ∈ V F 00(p)(v, v) = 0 ⇔ v ∈ span(p);

(ii) C-regular if it is C-regular at every point

p ∈ V \ {0};

(iii) a Finsler norm if it is C-regular and positive. Remark C-regular = 'regular in the sense of Caratheodory' (calculus of variations)

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C-REGULARITY Properties Let (1)

F

F

be a C-regular pre-Finsler norm. Then:

is subadditive, and hence it is convex.

(2) We have the fundamental inequality F 0(p)(v) ≤ F (v); p ∈ V \ {0}, v ∈ V.

(3) We have the Cauchy-Schwarz inequality |hu, viF | ≤ |F (u)||F (v)|; u ∈ V \ {0}, v ∈ V.

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C-REGULARITY If

F

is a Finsler norm, then

(4) Equality F (u + v) = F (u) + F (v) holds if and only if where λ is a non-negative real number.

u = λv

or

v = λu

(5) In the fundamental inequality we have equality if and only if λ ∈ R+.

v = λp,

(6) In Cauchy-Schwarz inequality we have equality, if and only if, λ ∈ R+.

v = λu,

(7)

F -orthogonality

is equivalent to Birkho-orthogonality.

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C-REGULARITY Proof of subadditivity

u ∈ V \ {0}, v ∈ V .

By Taylor's theorem

1 F (u + v) = F (u) + F 0(u)(v) + F 00(u + θ1v)(v, v), 2 1 F (u − v) = F (u) − F 0(u)(v) + F 00(u − θ2v)(v, v) 2 θ1, θ2 ∈ ]0, 1[. By the C-regularity of F , F (u + v) ≥ F (u) + F 0(u)(v), F (u − v) ≥ F (u) − F 0(u)(v)

whence 2F (u) ≤ F (u + v) + F (u − v). 1 (u − v) a := 1 (u + v), b := 2 2

F (a + b) ≤ F (a) + F (b)

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ELLIPTIC AND PSEUDO-FINSLER NORMS Denition A pre-Finsler norm (i) elliptic at a point

F :V →R

p ∈ V \ {0},

is

if the scalar product

gp := E 00(p) : V × V −→ R

is positive denite; (ii) elliptic if it is elliptic at every point of

V \ {0};

(iii) pseudo-Finsler if ³

´

³

00 (p) := F 00 (p) ¹ Ker F 0 (p) × Ker F 0 (p) F^

´

is a non-degenerate symmetric bilinear form at any point p ∈ V \ {0}, i.e., 00 (p)(u, e v e) = 0 for all if F^

ve ∈ Ker

³

´ 0 F (p)

then

e = 0. u

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ELLIPTIC AND PSEUDO-FINSLER NORMS Claim 3: If a pre-Finsler norm

F : V → R does not of F implies that F

Claim 4: If a pre-Finsler norm

F :V →R

of V \ {0} then the C-regularity norm.

vanish at any point is a pseudo-Finsler

does not vanish at any point 00 (p) is positive denite for of V \ {0} and the symmetric bilinear form F^ all p ∈ V \ {0}, then F is C-regular.

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CHARACTERIZATIONS OF FINSLER NORMS

For a positive pre-Finsler norm F : V → R the following are equivalent: (F1) F is C-regular, and hence it is a Finsler norm. (F2) F is elliptic. (F3) F is a pseudo-Finsler norm. (F4) At every point p ∈ V \ {0} and for any subspace H of V complementary to span(p), the symmetric bilinear form F 00(p) ¹ H × H

is non-degenerate.

(F5) F 00(p) ∈ L2sym(V ) has rank n − 1 at every point p ∈ V \ {0}. (F6) The scalar product gp ∈ L2sym(V ) is non-degenerate at every point p ∈ V \ {0}. Scheme of proof ⇒

⇒

(F 2) ⇐ (F 1) ⇐ (F 3) ⇒ (F 5) ⇒ ⇒ (F 6) (F 4) | {z }| {z } 1st cycle 2nd cycle

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PROOF OF (F1)⇒(F2) F :V →R

(F1) (F2) For all

F (

is a positive pre-Finsler norm

is C-regular , i.e.,

F 00(p)(v, v) ≥ 0 for all p ∈ V \ {0}, v ∈ V ; F 00(p)(v, v) = 0 ⇔ v ∈ span(p) F is elliptic , i.e., for all p ∈ V \ {0}, gp := E 00(p) ∈ L2 sym (V ) positive denite p ∈ V \ {0}, v ∈ V :

gp(v, v) = hp(v, v) +

³

´2 0 F (p)(v)

=

³ ´2 00 0 F (p)F {z (p)(v, v)} + F (p)(v) | {z } | ≥0 ≥0

⇒ gp(v, v) ≥ 0

and F 0(p)(v) = 0 (λ ∈ R) and F 0(p)(v) = 0 and F 0(p)(λp) = λF 0(p)(p) = λF (p) = 0

gp(v, v) = 0 ⇔ F 00(p)(v, v) = 0 ⇔ v = λp ⇔ v = λp ⇔ v=0

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F is a pseudo-Finsler norm , i.e., 00 (p) := F 00 (p) ¹ Ker(F 0 (p)) × Ker(F 0 (p)) F^

(F3)

is non-degenerate at all

(F1)

F

is C-regular

p ∈ V \ {0}

a Euclidean scalar product onoV Ã k k Euclidean norm, ⊥ Euclidean ¯ n orthogonality, SE := v ∈ V ¯¯kvk = 1 Euclidean unit sphere h, i

00 (p) does not depend on p ⇒ it is enough to show (1) The signature of F^ that there exists a point at which F is C-regular.

(2) (3)

F

achieves its minimum at a point

Ker

³

F 0(e)

´

³

´⊥

= span(e)

e ∈ SE .

Ã e⊥F v ⇔ e⊥v.

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PROOF OF (F3)⇒(F1)

(F3)

F

(F1)

F

(4)

is a pseudo-Finsler norm implies is a C-regular norm

v ∈ (span(e))⊥ \ {0} γ : R → V, t 7→ γ(t) := e + tv 1 γ : R→S c := kγk E F 0(e)

(5)

F ◦c:R→R

³

´ 00 c (0)

has a minimum at

= −kvk2F (e)

0∈R

and

(F ◦ c)00

exists

(4)

⇒

0 ≤ (F ◦ c)00(0) = F 00(e)(v, v) + F 0(e)(c00(0)) = F 00(e)(v, v) − kvk2F (e); ⇒

Another plan

F 00(e)(v, v) ≥ kvk2F (e) > 0.

The Euclidean norm square function f : V −→ R, u 7−→ f (e) := kuk2 = hu, ui

attains its maximum on the F -indicatrix SF =

n

¯ o ¯ v ∈ V ¯F (v) = 1 .

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PROOF OF (F6)⇒(F3) (F6)

gp = E 00(p) ∈ L2 sym (V ) p ∈ V \ {0}

(F3)

F

is non-degenerate at every point

is a pseudo-Finsler norm 00 (p) = F^

1 00 (p) gp ⇒ F^ F (p)

is non-degenerate

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CLOSING THE FIRST CYCLE The implication (F2) F is elliptic, i.e., ⇓

(F6)

gp

gp

is positive denite of all

is non-degenerate (p ∈ V

\ {0})

is evident, so we have: (F 2) ⇐ (F 1) ⇐ (F 3) ⇒ ⇒ (F 6)

p ∈ V \ {0}

28


(F3)


F is a pseudo-Finsler norm, i.e., F 00(p) ¹ Ker(F 0(p)) × Ker(F 0(p)) is

(F5) rank(F 00(p)) = n − 1 (p ∈ V (

non-degenerate (p ∈ V \ {0})

jF 00(p) : u ∈ V 7→ jF 00(p) ∈ V ∗, jF 00(p)(u)(v) := F 00(p)(u, v), v ∈ V. 0 (bi)n−1 i=1 a basis of Ker(F (p)) (Claim 1) b∗i := jF 00(p)(bi), (b∗i )n−1 i=1 is linearly independent

⇒

³

´ 00 rank F (p) ³ ´ 00 rank F (p)

≥n−1 ≤n−1

(Claim 2)

\ {0})

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PROOF OF (F5)⇒(F4) (F5) rank(F 00(p)) = n − 1 (p ∈ V \ {0}) (F4) For any subspace H of V complementary to span(p), F 00(p) ¹ H ×H is non-degenerate, i.e., if u ∈ H and F 00(p)(u, v) = 0 for all v ∈ H , then u = 0. (1) rank(F 00(p)) := rank(jF 00(p)) := dim Im(jF 00(p)) cond. = n−1 (b∗i )n−1 i=1 (bi)n−1 i=1

a basis of Im(jF 00(p)) is dened by b∗i = jF 00(p)(bi), i ∈ {1, . . . , n − 1}.

Then V.

(bi)n−1 i−1

is linearly independent and

(b1, . . . , bn−1, p)

is a basis of

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PROOF OF (F5)⇒(F4) (F5) rank(F 00(p)) = n − 1 (p ∈ V \ {0}) (F4) For any subspace H of V complementary to F 00(p) ¹ H × H is non-degenerate. (2) If

F 00(p)(u, v) = 0

for all

v ∈ H,

then

span(p),

F 00(p)(u, w) = 0

for all

w ∈V.

w = v + λp; v ∈ H, λ ∈ R F 00(p)(u, w) = F 00(p)(u, v) + λF 00(p)(u, p) = 0 + 0

(3)

u=

Pn−1 i=1 λi bi + λp.

we have (2)

0 = F 00(p)(u, w) =

If

F 00(p)(u, v) = 0

n−1 X

⇒

i=1

(since

v ∈ H,

then for all 

λiF 00(p)(bi, w) + λF 00(p)(p, w) = 

i=1 n−1 X

for all

n−1 X i=1

w∈V 

λib∗i  (w)

λib∗i = 0 ∈ V ∗ ⇒ λ1 = · · · = λn−1 = 0 ⇒ u = λp ⇒ u = 0

H ∩ span(p) = {0}).

31

CLOSING THE SECOND CYCLE (F4)⇒(F3) evident, hence: ⇒

(F 5) ⇒

⇒

(F 3)

(F 4)

32

CRAMPIN'S THEOREM If F : V → R is a symmetric C-regular pre-Finsler norm, then positive, hence it is a Finsler norm.

F

is

Step 2: If F vanishes at a point of V \ {0}, then there is a point where is positive. (Taylor's formula)

F

Sketch of proof: Step 1:

F

cannot be everywhere negative on

Step 3: There is no point in + symmetry of F ) Step 4: There is a point in

V \ {0}

V \ {0}

Step 5: There is no point in value theorem.)

where

where

V \ {0}

where

vanishes. (Taylor's formula

F

F

V \ {0}.

is positive. F

is negative. (Intermediate