Convex bodies with homothetic sections.

8/11/2019 Convex bodies with homothetic sections.

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CONVEX BODIES WITH HOMOTHETIC SECTIONS

L U I S M O N T E J A N O

ABSTRACT

We prove that if

K

is a convex body in E

n+ 1

,

n

^ 2, and

p

0

is a point of

K

with the property that all bi-

sections of K through p

0

are homothetic, then K is a Euclidean ball.

1. Introduction

Let

AT

be a convex body in E"

+1

,

n

^ 2, and let

p

0

be a point of

K.

Suppose that all -sections of

K

through

p

0

are affinely equivalent. Must

K

be an

ellipsoid? If

n

is even, Gromov [4] and independently Mani [5] and Burton [2] proved

that the answer is yes. The problem is still unsolved when n is odd.

Suppose now that all -sections of

K

through

p

0

are congruent. If

n

is even,

Gromov's result implies that K is an ellipsoid and hence it is easy to check that K is

a Euclidean ball; if

n

= 3, Burton [2] proved that

K

is a Euclidean ball; and finally,

Schneider [7] gave a proo f of the same sta tem ent for n ^ 2. The purpo se of this paper

is to prove the following results.

THEOREM

1.

If all n-sections of K through p

0

are affinely equivalent, then either K

is an ellipsoid or K is centrally symmetric with respect to p

0

.

THEOREM 2.

If all n-sections of K through p

0

are volume-preserving affinely

equivalent, then K is a Euclidean ball.

As a corollary, we have Schneider's Theorem.

COROLLARY. If all n-sections ofK through p

0

are congruent, then K isa Euclidean

ball.

THEOREM

3.

If all n-sections ofK through p

0

are homothetic, then K

is

a Euclidean

ball.

The corresponding results about ^-sections, 2 ^

k

^

n,

follow immediately from

the above results or their proofs.

2. Definitions and preliminaries

Let 7

n +1

be the group of isometries of E

n+ 1

, the (n+l)-dimensional Euclidean

space, and let

O

n+ 1


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382 LUIS MONTOANO

by

S

k

~

l

the standa rd unit sphere in E*, and by

O

k

the subgroup of

O

n+ 1

containing

all elem ents w hich carry E* on to itself and a ct as the identity o n the sub spac e of E

n+1

orthogonal to E*. For every

xeS

n

,

let

E(x)

be the tangent plane of

S

n

at

x

and let

H(x) be the ^-dimensional subspace of E

n+ 1

orthogonal to x. Hence, H(x) = H{x)

and

E(x) = x + H(x).

Finally, let

B

n

= {Y

d

i-i

t

i

x

i

eSn

\

t

n+i >

)

b e t h e

topological

n-

ball contained in S

n

with boundary

5"

1

"

1

.

By a convex body in E

n+1

we understand a non-empty compact convex subset of

E

n+1

. Let us denote by Q the space of all convex bodies in E

n+ 1

with the topology

induced by the Hausdorff metric. We consider the (continuous) map x:Q -* E

n+1

which assigns to each C e Q the centre of its circumscribed sphe re. Let C

1

and

C

2

eQ. We say that C

x

is congruent to C

2

if there is gel

n+ 1

such that g(C

x

) = C

2

.

Fur thermore, C

x

ishomothetic to C

2

if X{C

X

) is congruent to C

2

for some AeR{0}.

From now on, let C O

n+ l

/G

0

and n:O

n+ 1

/G

0

-> O

n+ 1

/O

n

th e

canonical projections. Note that II and n are fibre bundles with fibres G

o

and O

n

/G

0

,

respectively. Let C

o

=

x

n+ 1

+ C a E(x

n+ 1

)

an d let T

o

=

{g(C

0

)eQ\geO

n+ 1

}

with the

topology induced by the Hausdorff metric. That is, T

o

is the space of all

n-

dimensional convex bodies congruent to C and tangent to

S

n

at their circumcentres.

Let us consider the following diagram:

n n

O

n+l

>O

n+ l

/G

0

>O

n+ 1

/O

n

11 ^ |

siv

where

T(g)

=

g(C

0

)

for every

geO

n+ 1

, V(gG

0

) = g(C

0

)

for every

g

G

0

eO

n+ 1

/G

0

and

y/(gO

n

) = g(x

n+ 1

)

for every

gO

n

e

O

n+

JO

n

. It is no t difficult to pro ve tha t

the functions described above are well-defined continuous maps, the diagram

is com mu tative, and the map s *F and y/ are hom eom orphisms. Consequently,

T: O

n+ 1

-> T

o

and

T: T

O

->

S

n

are fibre bundles with fibres

G

o

and

OJG

Q

,

respectively.

Note also that for every geO

n+ 1

, xT{g) = g(x

n+ 1

).

Afield of convex bodies congruent to C tangent to S

n

is a map

which is a section of

T : T

0

-*S

n

(that is,

XK =

Id

s

n) and also has the property that

A complete turning of C in

E

n+1

K

:S

n

To

is a field of convex bodies congruent to

C

tangent to

S

n

such that

K(X) x = K( x) + x

or, equivalently, such that

I

X

K(X)

= K(-X),

where i

x

e0

n+l

is the reflection across H(x).

If

n

is even, Mani [5] proved that the existence of a field of convex bodies

congruent to C tangent to S

n

implies that C is a Euclidean ball. On the other hand,

Bu rton [2] proved that there is a com plete turning of C in E

4

if and only if

C

is


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CONVEX

BODIES WITH HOMOTHETIC SECTIONS

3 8 3

centrally symmetric. Furthermore, it is not difficult to see that if all /i-sections of

K

through the origin are congruent (respectively affinely equivalent), then there is a

complete turning of a congruent (respectively affinely equivalent) copy of

K n

E

n

in

E

n+ 1

.

3. The proofs of the theorems

We shall start by proving the following lemma.

LEMMA 1.

Let K:S

n

-> Y

o

be afield of convex bodies congruent to C tangent to S

n

.

Then there is a map

K(B

H

) is a trivial fibre bund le. Therefore , th ere is a m ap

4>-.K(B

n

)->0

n+ 1

such that

T4> = Id

K( B

n

)

and 0(C

O

) = l e O

n + 1

. Let

O

n+ 1

be

the composition

O

n

as follows:


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384 LUIS MONTEJANO

Note that since C

4

e(j

0

, C,(JC

0

) # -x

0

, and therefore

(t,s)

iswell-defined bec ause

Note also that for every

seS

n

-\

(0,s) =

1

eO

n

and

Define the map N : 5

n

- O

n+1

as follows: letx = Yji-i

l

i

x

i

eBn

^

t n e n

Clearly,

ATis a

well-defined continuous

map.

M oreover ,

we

shall prove that

N

satisfies

the two

condit ions

(a )

N

x

(x

0

)eH(x), for every XGB

71

, and

(b) N() ^ W

+ U-{0)

such that for every xsS

n

,

K

(x) = (Kf)H(x))/X(x) +x.

Consequently, since d(C)= 1, we have that for every xeS

n

,

\X(x)\ = d(K0H(x)),

and sinceX x

n+1

) = 1, by continuity, we have that for every

xeS

n

,

X x) = d(K(]H(x)).

Let I""

1

be the unit sphere ofH(x

n

); that is, I""

1

= H(x

n

) nS

n

= S

Xn

.Note that

Z""

1

is the boundary of a topological ball contained in

S

n

.

Note also that for every

s e l " "

1

, x

n

eH(s), and therefore d(K(]H(s)) = 1. Consequently,

X s) =1, for every seE ""

1

.

CLAIM.

Let Y be any

{n\)-dimensional subspace

ofH(x

n

). Then d(K()

F) = 1.

Without loss of generality, we may choose coordinates in such a way that

F = E""

1

. Suppose that

d(K(]

E

n-1

) < 1. As in Lemma 1, since S""

1

is the boundary

of a topological n-ball contained in 5

n

, there is a map

O : " "

1

>O

n 1

such that

14 BLM23


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3 8 6 LUIS MONTEJANO

(a) O,(C

0

) =

K

(s) = K()H(s) +s,

for every

(b) O,(x

n+1

) = s, for every seZ""

1

, and

We shall prove first that for every seZ""

1

, O

s

(x

n

)^ / / ( jc

n

). Suppose that there is

^ e l " -

1

such that

O,JLxJeH(xJ.

By (b), O

8o

(x

n+ 1

) =

s

o

eH(x

n

),

hence x . e O . / E -

1

)

and therefore

/s: n {/*

11

e u} c A : n O ^ E * -

1

=

Consequently, d(K(] E

71

"

1

) = 1, which is a contradiction.

Note now that for every seZ""

1

,

s

(jc

n

)

^ x

n

otherwise, since O

n + 1

as follows:

#

s

= A(O

s

(x

r t

) ,x

n

)O

s

, for every

s

el"-

1

.

N ot e t ha t JV is well-defined beca use

O,(JC

B

)

# x

n

. Note also that

N

s

(x

n+ 1

)

= 5,

because 5 is orthogon al to bo th x

n

and

Q>

s

(x

n

).

Fur thermore ,

N

s

(x

n

)

= jc

n

.

Consequent ly , ^(E""

1

) is orthogonal to both s and x

n

and hence, for every

Consequently, for every SET,

71

'

1

we have the equality

N

s

(x

n

+ E""

1

) = ta ngen t ( - l)-pla ne ofZ""

1

at 5,

which gives a trivialization of the tangent space of the (w-l)-sphere Z

n

~\ This is a

contradiction becausenis odd (see 27.5 of [8]). With this we conclude the proof of the

claim.

Observe now that for every n-dimensional subspace

H{x)

of E

n+ 1

,

d(K 0 H(x))

=

d{K) =

1, because if

T

=

H(x)

n

H(x

n

),

then by the above claim, 1 =

d(Kf] T)

^

d(K

n

H(x))

^ f/(AT) = 1. The ref ore , all ^-sectio ns of

K

through the origin are

congruent (that is,

X{x) =

1 for every

xeS

n

)

and hence, by the corollary of Th eorem

2,

K

is a Euclidean ball.

References

1. P. W.AITCHISON, C. M. PETTY and C. A. ROGERS, 'A convex body w ith a false centre is an ellipsoid',

Mathematika 18 (1971) 50-59 .

2. G. R. BURTON, 'Congr uent sections of a convex b ody ', Pacific J. Math. 81 (1979) 303-316.

3. L. DANZER, D . LAUGWITZ and H. LENZ, 'Uber das Lownersche Ellipsoid und sein Analogen unter den

einem Eikorpe r einbeschreibenen Ellipsoide n', Arch. Math. 8 (1957) 214-219.

4. M. L.GROMOV, 'On a geometric hypothesis of Banach' (Russian),

Izv.

Akad.

Nauk SSSR, Ser. Mat.

31

(1967) 1105-1114; MR 35, No. 655.

5. P. MANI, 'Fields of planar bodies tangent to spheres', Monatsh. Math. 74 (1970) 145-14 9.

6. J. W. MILNOR and J. D. STASHEFF, Characteristic classes, Ann. of Math. Studies 76 (Princeton

University Press, 1974).

7. R. SCHNEIDER, 'Convex bodies with congruent sections',

Bull.

London M ath. Soc. 12 (1980) 52-54.

8. N. STEENROD, The topology of fiber bundles (Princ eton Unive rsity Press, 1951).

Insti tuto de Matematicas

Universidad Nacional Autonoma de Mexico

Ciudad Universitaria, Circuito Exterior

Mexico D.F., 04510

Mexico

Convex bodies with homothetic sections.

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