On convex bodies of constant width
Abstract We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in R n is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.
Structure of the Space of Diametrically Complete Sets in a Minkowski Space
We study the structure of the space of diametrically complete sets in a finite dimensional normed space. In contrast to the Euclidean case, this space is in general not convex. We show that its starshapedness is equivalent to the completeness of the parallel bodies of complete sets, a property studied in Moreno and Schneider (Isr. J. Math. 2012, doi:10.1007/s11856-012-0003-6), which is generically not satisfied. The space in question is, however, always contractible. Our main result states that in the case of a polyhedral norm, the space of translation classes of diametrically complete sets of given diameter is a finite polytopal complex. The proof makes use of a diagram technique, due to P. McMullen, for the representation of translation classes of polytopes with given normal vectors. The paper concludes with a study of the extreme diametrically complete sets.
Ultrastructural observations of the spermiogenesis in the Parotoplaninae (Plathelminthes, Proseriata) with special reference to a striated appendage of the intercentriolar body
SummaryThe spermiogenesis of three species of the Parotoplaninae (Otoplanidae, Proseriata) is described based on electron-microscopical observations. Special reference is given to organelles which do not persist in mature male gametes. One of these organelles is a striated appendage of the intercentriolar body. This differentiation has not been reported from any other plathelminth taxa up to now. The striated appendage, which may serve as a strengthening element, is hypothesized to be an autapomorphic feature of the Parotoplaninae.