Download
1 / 36
360 likes | 374 Views
This paper explores inequalities describing geometric properties of chords halving the area of a planar bounded convex set. It delves into definitions such as halving partners, breadth measures, and v-halving distance. The results include propositions, lemmas, and a conjecture on the maximum v-halving distance optimal for planar convex sets. The study connects the concept of halving chords to fencing problems and poses open questions regarding the structure of planar convex sets. The text also references related works, offering a comprehensive view of the topic.
E N D
Chords halving the area of a planar convex set Grüne , E. Martínez, C. Miori, S. Segura Gomis
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.
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.
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:
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.
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 ofProposition 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 . . .
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), 464-476.
