Chords halving the area of a planar convex set A.Grüne, E. Martínez, C. Miori, S. Segura Gomis

Chords halving the area of a planar convex set

A. Grüne , E. Martínez, C. Miori,

S. Segura Gomis

1.Introduction

Problem: to determine some inequalities describing geometric properties of the chords halving the area of a planar bounded convex set K.

- A. Ebbers-Baumann, A. Grüne, R. Klein: Geometric dilation of closed planar curves: New lower bounds. To appear in Theory and Applications dedicated to Euro-CG ’04, 2004.

2. Definitions

2.1 Halving partner.

Let K be a planar convex set.

Let p be a point on .

Then the unique halving partner p' on

is the intersection point between the straight line pp' halving the area of K and its boundary.

KK

2. Definitions

2.2 Breadth measures.

. v-length :

vpqpqKqppqvKl ,,:max:),(

2. Definitions


. diameter :

),(max:)(1

vKlKDSv

2. Definitions


. minimal width :

),(min:)(1

vKlKSv

2. Definitions


. v-breadth :

vpvpvKbKp

,min,max:),(

2. Definitions

2.3 v-halving distance:

is the distance of the halving pair with direction v.

),(max:)(1

vKhKH ASv

A

),(min:)(1

vKhKh ASv

A

),( vKhA

Proposition 1: )()()()( KhKKHKD AA

Proof of Proposition 1:

1. it is trivial.

2.

Rotating v in there is at least, by continuity, a direction v0 such that the maximal chord in this direction divides K into two subsets of equal area. Then:

3. For every v, Then:

)()( KKHA )()( KHKD A

1S

)(),(),()( 00 KHvKhvKlK A

),(),( vKhvKl A

)(),(min),(min11

KhvKhvKl AASvSv

3. Overview of the results

ω D p r R A

AH none DH A pH A 21

none RH A 2 none

AH rH A 2

Ah Ah DhA phA 1

RhA 2

Ah none none none none


ω D p r R A

AH none DH A pH A 21

none RH A 2 none

AH rH A 2

Ah Ah DhA phA 1

RhA 2

Ah none none none none

CS

CS ?, ?,


ω D p r R A

AH n o n e DH A pH A 21

n o n e RH A 2 n o n e

AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1

Rh A 2

Ah 2

1Ah n o n e n o n e rh A 2 n o n e n o n e

CS

CS

?, ?,


ω D p r R A



AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1

Rh A 2

Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

pH A 43

rH A 2

RH A 433

AH A

42

Ah Ah Dh A ph A 1

rh A 3 Rh A 2

Ah A 32

Ah 2


CS

CS

?, ?,

EC ,

Lemma 1 (Kubota):

If is a convex body, then

Lemma 2 (Grüne , Martínez, – – , Segura) :

If is a convex body, then

This bound cannot be improved.

2K

2DA

2K

DhA A

Lemma 1 + Lemma 2

2

Ah

Lemma 1 + Lemma 2

2

Ah

2

Ah

Proposition 2:

If is a convex body, then .


Lemma 3:

If is a convex body, and is an arbitrary direction, then .


2K 2Ah

2K 1Sv2lhA

Proof of the Lemma 3:

ppvhA )(

qqvl )(

21 llc

rightleft AA

)()( 22 vh

A

vl

AA

A

rightrightleft

21

)()(

2

2

rightright

right

rightleft

rightA

AAA

AAA

vlvh

Proof of Proposition 2:

Let be the direction such that

Then we get:

1Sv )(vhh AA

22

)()(

3

vlvhh

Lemma

AA

Proposition 3:

For any convex body K we have

This bound is tight.

DHA 43

Proof of the Proposition 3:

. D = pq

.

qp ,

Assume

.

.

.

bottomtop AA

rightleft AA

343

DHA

1,1

yy qp

rightleft AA

xx

xxx

qp

qpb

34

x

xx

xxx b

qp

qpc 34

Contradiction!


ω D p r R A



AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1

Rh A 2

Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1

Rh A 2

Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1

rh A 3

Rh A 2

Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1

rh A 3 Rh A 2 Ah A 32

Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1


Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

pH A 43

rH A 2 AH A

42

Ah Ah Dh A ph A 1


Ah 2


CS

CS

?, ?,

EC ,


ω D p r R A



AH AH DH A 43

pH A 43

rH A 2 RH A 433

AH A 42

Ah Ah Dh A ph A 1


Ah 2


CS

CS

?, ?,

EC ,

4. Conjecture and open problems4.1 In the family of all bounded convex sets

where the maximum is attained if and only if K is a disc. The conjecture was first posed by Santaló. The best bound known up to now, which is a consequence of Pal’s Theorem, is

...12838.12

)(

)(

KA

KhA

...31607.13)(

)( 4 KA

KhA

4. Conjecture and open problems4.2 Are discs the only planar convex sets with constant v-halving distance? Equivalently, is the lower bound of the ratio

attained ONLY by a disc?

1)()(

KhKH

A

A

5. Final remark

The chords halving the area of a planar bounded convex set are involved in the so called fencing problems which consider the best way to divide by a “fence” such sets into two subsets of equal area.

5. Final remark

The chords halving the area of a planar bounded convex set are involved in the so called fencing problems which consider the best way to divide by a “fence” such sets into two subsets of equal area.

- H. T. Croft, K. J. Falconer, R. K. Guy: Unsolved problems in Geometry. Springer-Verlag, New York (1991), A26;

- C.M, C. Peri, S. Segura Gomis: On fencing problems, J. Math. Anal. Appl. (2004), 464-476.

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Chords halving the area of a planar convex set A.Grüne, E. Martínez, C. Miori, S. Segura Gomis