Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space M ^d is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M ^d . We establish a number of properties of reduced bodies in M ². They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M ² is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.     

Reduced convex bodies in Euclidean space—A survey

A convex body R in Euclidean space E^d is called reduced if the minimal width Δ(K) of each convex body K⊂R different from R is smaller than Δ(R). This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in E² are regular odd-gons. We know a relatively large amount on reduced convex bodies in E². Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d≥3 we do not even have tools which permit us to recognize what the boundary of R looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in E^d, recall some striking related research problems, and put a few new questions.     

Spherical bodies of constant width

The intersection L of two different non-opposite hemispheres G and H of the d-dimensional unit sphere \(S^d\) is called a lune. By the thickness of L we mean the distance of the centers of the \((d-1)\)-dimensional hemispheres bounding L. For a hemisphere G supporting a convex body \(C \subset S^d\) we define \(\mathrm{width}_G(C)\) as the thickness of the narrowest lune or lunes of the form \(G \cap H\) containing C. If \(\mathrm{width}_G(C) =w\) for every hemisphere G supporting C, we say that C is a body of constant width w. We present properties of these bodies. In particular, we prove that the diameter of any spherical body C of constant width w on \(S^d\) is w, and that if \(w < \frac{\pi }{2}\), then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide.     

When is a spherical body of constant diameter of constant width?

Aequationes mathematicae - We prove that a smooth convex body of diameter $$\delta &lt; \frac{\pi }{2}$$ on the d-dimensional unit sphere $$S^d$$ is of constant diameter $$\delta $$ if and only...     

Reduced bodies in Minkowski space

Summary We study the existence of almost-periodic solutions for a second order abstract Cauchy problem defined in a Banach space.     

On-line covering the unit cube by cubes

We consider two on-line methods of covering the unit cube of Euclideand-space by sequences of cubes. The on-line restriction means that we are given the next cube from the sequence only after the preceding cube has been put in place without the possibility of changing the placement. The first method enables on-line covering of the unit cube by an arbitrary sequence of cubes whose total volume is at least 3...2 ^d −4. The second method is more complicated, but, asymptotically, asd tends to infinity, it yields an efficiency of the order of magnitude 2 ^d with factor 1. So, asymptotically, it is as good as the best possible non-on-line method of covering the unit cube by cubes. This research was supported in part by Komitet Badań Naukowych (Committee of Scientific Research), Grant Number 2 2005 92 03.     

A β-hairpin epitope as novel structural requirement for protein arginine rhamnosylation

For canonical asparagine glycosylation, the primary amino acid sequence that directs glycosylation at specific asparagine residues is well-established. Here we reveal that a recently discovered bacterial enzyme EarP, that transfers rhamnose to a specific arginine residue in its acceptor protein EF-P, specifically recognizes a β-hairpin loop. Notably, while the in vitro rhamnosyltransferase activity of EarP is abolished when presented with linear substrate peptide sequences derived from EF-P, the enzyme readily glycosylates the same sequence in a cyclized β-hairpin mimic. Additional studies with other substrate-mimicking cyclic peptides revealed that EarP activity is sensitive to the method used to induce cyclization and in some cases is tolerant to amino acid sequence variation. Using detailed NMR approaches, we established that the active peptide substrates all share some degree of β-hairpin formation, and therefore conclude that the β-hairpin epitope is the major determinant of arginine-rhamnosylation by EarP. Our findings add a novel recognition motif to the existing knowledge on substrate specificity of protein glycosylation, and are expected to guide future identifications of rhamnosylation sites in other protein substrates.

For bacterial arginine rhamnosylation, the rhamnosyltransferase EarP specifically recognizes a β-hairpin structure in the acceptor substrate.     

On the smallest disk containing a planar reduced convex body

A convex body R of Euclidean space E ^d is said to be reduced if every convex body $ P \subset R $ different from R has thickness smaller than the thickness $ \Delta(R) $ of R. We prove that every planar reduced body R is contained in a disk of radius $ {1\over 2}\sqrt 2 \cdot \Delta(R) $. For $ d \geq 3 $, an analogous property is not true because we can construct reduced bodies of thickness 1 and of arbitrarily large diameter.     

Approximation of convex bodies by rectangles

For every plane convex body there is a pair of inscribed and circumscribed homothetic rectangles. The positive ratio of homothety is not greater than 2.Research supported in part by Komitet Badan Naukowych (Committee of Scientific Research), grant number 2 2005 92 03.