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 halvingthe 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 v0such that the maximal chord in this direction divides K into two subsets of equal area. Then: • For every v, Then:

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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 ofProposition 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!

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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 Pal’s 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.