Strong Klee-And\^o Theorems through an Open Mapping Theorem for cone-valued multi-functions

A version of the classical Klee-And\^o Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $\alpha>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$. The conclusion of the Klee-And\^o Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-And\^o Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-And\^o Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-And\^o Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space $X$, ordered by a closed generating cone $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$.Comment: Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier method

Normality of spaces of operators and quasi-lattices

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that are not Banach lattices, but for which $B(X,Y)$ is normal. In particular, we show that a Hilbert space $\mathcal{H}$ endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and satisfies an identity analogous to the elementary Banach lattice identity $\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.Comment: Minor typos fixed. Exact solution now provided in Example 5.10. To appear in Positivit

On compact packings of the plane with circles of three radii

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$ We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ for which there exist a compact circle-packing of the plane consisting of circles with radii $s$, $r$ and $1$. We discuss computing the exact values of such $0<s<r<1$ as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing \emph{all} these values on contemporary consumer hardware with the methods employed in this paper.Comment: Dataset referred to in the text can be obtained at http://dx.doi.org/10.17632/t66sfkn5tn.

Geometric duality theory of cones in dual pairs of vector spaces

This paper will generalize what may be termed the "geometric duality theory" of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual Banach space. We show that geometric duality theory is not restricted to real pre-ordered Banach spaces, as is done classically, but can be extended to real Banach spaces endowed with arbitrary collections of closed cones. We define geometric notions of normality, conormality, additivity and coadditivity for members of dual pairs of real vector spaces as certain possible interactions between two cones and two convex convex sets containing zero. We show that, thus defined, these notions are dual to each other under certain conditions, i.e., for a dual pair of real vector spaces $(Y,Z)$, the space $Y$ is normal (additive) if and only if its dual $Z$ is conormal (coadditive) and vice versa. These results are set up in a manner so as to provide a framework to prove results in the geometric duality theory of cones in real Banach spaces. As an example of using this framework, we generalize classical duality results for real Banach spaces pre-ordered by a single closed cone, to real Banach spaces endowed with an arbitrary collections of closed cones. As an application, we analyze some of the geometric properties of naturally occurring cones in C*-algebras and their duals

Modulation of lipid mediator pathways by Staphylococcus aureus in human macrophages

Staphylococcus aureus (S. aureus) causes serious infectious diseases. To develop new treatments against S. aureus infections in-depth knowledge of S. aureus-host interactions is required. The modulation of lipid mediator (LM) pathways by S. aureus remained largely unexplored so far. We show that S. aureus modulates LM formation in a model of osteomyelitis in vivo as well as in osteoclasts and monocyte-derived macrophages (MDM) in vitro. In both murine osteoclasts and human MDM S. aureus exposure caused an elevation of cyclooxygenase (COX)-2 and microsomal prostaglandin E synthase-1 (mPGES-1) expression along with increased prostaglandin E2 formation. In human MDM challenged with S. aureus during polarization, the expression of 15-lipoxygenase (LOX)-1 was prevented, which resulted in impaired formation of 15 LOX-derived LMs. The S. aureus challenge increased the expression of M1 surface markers, while it reduced M2 surface marker expression, indicating that S. aureus shifts M2- towards M1-like MDM. Our results suggest that lipoteichoic acid (LTA) largely accounts for the S. aureus-induced LM pathway modulation. We show that the induction of COX 2 and mPGES 1 expression by LTA is mediated via signaling pathways involving mainly NF κB and p38 MAPK. Impairment of 15-LOX-1 expression by LTA correlates to reduced Lamtor1 expression. We report that S. aureus even alters established phenotypes of fully polarized M1 and M2. Besides, we uncover that the SCV strain JB1 releases less LTA during growth than the wild-type strain 6850, whereas both strains contain comparable amounts of LTA. Our data suggest that COX-derived LMs may promote the survival of S. aureus. Finally, we present that strain JB1 scarcely elicits LM formation in MDM upon short-term exposure. Together, this thesis reveals how LM pathways are modulated by S. aureus. Our data add to the characterization of S. aureus SCVs. Thereby, our findings advance the current knowledge on S. aureus-host interactions