Chords halving the area of a planar convex set

1
Chords halving the area of a planar convex set

• Grüne , E. Martínez, C. Miori,
• S. Segura Gomis

2
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.

3
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.

4
2. Definitions

• 2.2 Breadth measures.
• . v-length

5
2. Definitions

• 2.2 Breadth measures.
• . diameter

6
2. Definitions

• 2.2 Breadth measures.
• . minimal width

7
2. Definitions

• 2.2 Breadth measures.
• . v-breadth

8
2. Definitions

• 2.3 v-halving distance
• is the distance of the halving pair with
  direction v.

9

• Proposition 1

10

• Proof of Proposition 1
• it is trivial.
• 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
• For every v, Then

11
3. Overview of the results
 
12
3. Overview of the results
 
13
3. Overview of the results
 
14
3. Overview of the results
 
15
3. Overview of the results
 
16

• 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.

17
Lemma 1 Lemma 2
18
Lemma 1 Lemma 2
19

• Proposition 2
• If is a convex body, then
  .
• This bound cannot be improved.
• Lemma 3
• If is a convex body, and
  is an arbitrary direction, then
  .
• This bound cannot be improved.

20

• Proof of the Lemma 3

21

• Proof of Proposition 2
• Let be the direction such that
• Then we get

22

• Proposition 3
• For any convex body K we have
• This bound is tight.

23

• Proof of the Proposition 3
• . D pq
• .

24

• Assume
• .
• .
• .

25

Contradiction!
26
3. Overview of the results
 
27
3. Overview of the results
 
28
3. Overview of the results
 
29
3. Overview of the results
 
30
3. Overview of the results
 
31
3. Overview of the results
 
32
3. Overview of the results
 
33
4. Conjecture and open problems
4.1 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 Pals Theorem, is
34
4. Conjecture and open problems
4.2 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?
35
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.

36
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.