A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

Variations of the high-level Balmer line spectrum of the helium-strong star Sigma Orionis E

Using the high-level Balmer lines and continuum, we trace the density structure of two magnetospheric disk segments of the prototypical Bp star sigma Ori E (B2p) as these segments occult portions of the star during the rotational cycle. High-resolution spectra of the Balmer lines >H9 and Balmer edge were obtained on seven nights in January-February 2007 at an average sampling of 0.01 cycles. We measured equivalent width variations due to the star occultations by two disk segments 0.4 cycles apart and constructed differential spectra of the migrations of the corresponding absorptions across the Balmer line profiles. We first estimated the rotational and magnetic obliquity angles. We then simulated the observed Balmer jump variation using the model atmosphere codes synspec/circus and evaluated the disk geometry and gas thermodynamics. We find that the two occultations are caused by two disk segments. The first of these transits quickly, indicating that the segment resides in a range of distances, perhaps 2.5-6R_star, from the star. The second consists of a more slowly moving segment situated closer to the surface and causing two semi-resolved absorbing maxima. During its transit this segment brushes across the star's "lower" limb. Judging from the line visibility up to H23-H24 during the occultations, both disk segments have mean densities near 10^{12} cm^{-3} and are opaque in the lines and continuum. They have semiheights less than 1/2 of a stellar radius, and their temperatures are near 10500K and 12000K, respectively. In all, the disks of Bp stars have a much more complicated geometry than has been anticipated, as evidenced by their (sometimes) non-coplanarity, de-centerness, and from star to star, differences in disk height.Comment: Accepted by Astron. Astrophys, 13 pages, 4 embedded figure