Kalai's squeezed 3-spheres are polytopal

In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit.Comment: 11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometr

The Erd\H{o}s-Szekeres problem for non-crossing convex sets

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies

Geometry of the Prytz Planimeter

The Prytz planimeter is a simple example of a system governed by a non-holonomic constraint. It is unique among planimeters in that it measures something more subtle than area, combining the area, centroid and other moments of the region being measured, with weights depending on the length of the planimeter. As a tool for measuring area, it is most accurate for regions that are small relative to its length. The configuration space of the planimeter is a non-principal circle bundle acted on by SU(1,1), (isom. to SL(2,R)). The motion of the planimeter is realized as parallel translation for a connection on this bundle and for a connection on a principal SU(1,1)-bundle. The holonomy group is SU(1,1). As a consequence, the planimeter is an example of a system with a phase shift on the circle that is not a simple rotation. There is a qualitative difference in the holonomy when tracing large regions as opposed to small ones. Generic elements of SU(1,1) act on S^1 with two fixed points or with no fixed points. When tracing small regions, the holonomy acts without fixed points. Menzin's conjecture states (roughly) that if a planimeter of length L traces the boundary of a region with area A > pi L^2, then it exhibits an asymptotic behavior and the holonomy acts with two fixed points, one attracting and one repelling. This is obvious if the region is a disk, and intuitively plausible if the region is convex and A >> pi L^2. A proof of this conjecture is given for a special case, and the conjecture is shown to imply the isoperimetric inequality.Comment: AmS-TeX, 23 pages, 12 figures in 2 *.gif files. To appear in Reports on Mathematical Physics. Part of proceedings of Workshop on Non-holonomic Constraints in Dynamics, Univ. of Calgary, Aug. 199

A note on a class of pp-valent starlike functions of order beta

In this paper we obtain sharp coefficient bounds for certain $p$-valent starlike functions of order $\beta$, $0\le \beta<1$. Initially this problem was handled by Aouf in "M. K. Aouf, On a class of $p$-valent starlike functions of order $\alpha$, Internat. J. Math. $\&$ Math. Sci. 1987;10:733--744". We pointed out that the proof given by Aouf was incorrect and a correct proof is presented in this paper.Comment: 6 pages, 1 table, submitted to a journa