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.

2. Definitions 2.2 Breadth measures.. v-length :

2. Definitions 2.2 Breadth measures.. diameter :

2. Definitions 2.2 Breadth measures.. minimal width :

2. Definitions 2.2 Breadth measures.. v-breadth :

2. Definitions 2.3 v-halving distance: is the distance of the halving pair with direction v.

Proposition 1:

Proof of Proposition 1: 1. it is trivial. 2. Rotating v in there is at least, by continuity, a direction v 0 such that the maximal chord in this direction divides K into two subsets of equal area. Then: 3.For every v, Then:

3. Overview of the results

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.

Lemma 1 + Lemma 2

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.

Proof of the Lemma 3:

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

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

Proof of the Proposition 3:. D = pq.

Assume.

Contradiction!

3. Overview of the results

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

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?

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),