An Optimal Algorithm for the Separating Common Tangents of two Polygons

We describe an algorithm for computing the separating common tangents of two simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies to the same side of the line. A separating common tangent of two polygons is a tangent of both polygons where the polygons are lying on different sides of the tangent. Each polygon is given as a read-only array of its corners. If a separating common tangent does not exist, the algorithm reports that. Otherwise, two corners defining a separating common tangent are returned. The algorithm is simple and implies an optimal algorithm for deciding if the convex hulls of two polygons are disjoint or not. This was not known to be possible in linear time and constant workspace prior to this paper. An outer common tangent is a tangent of both polygons where the polygons are on the same side of the tangent. In the case where the convex hulls of the polygons are disjoint, we give an algorithm for computing the outer common tangents in linear time using constant workspace.Comment: 12 pages, 6 figures. A preliminary version of this paper appeared at SoCG 201

Diagrams as Vehicles for Scientific Reasoning

We argue that diagrams are not just a communicative tool but play important roles in the reasoning of biologists: in characterizing the phenomenon to be explained, identifying explanatory relations, and developing an account of the responsible mechanism. In the first two tasks diagrams facilitate applying visual processing to the detection of patterns that constitute phenomena or explanatory relations. Diagrams of a mechanism serve to guide reasoning about what parts and operations are needed and how potential parts of the mechanism are related to each other. Further they guide the development of computational models used to determine how the mechanism will behave. We illustrate each of these uses of diagrams with examples from research on circadian rhythm

Diameter two properties, convexity and smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.Comment: Removed Proposition 2.7 from version [v1] because of a gap in the proof. arXiv admin note: text overlap with arXiv:1506.0523

Strongly extreme points and approximation properties

We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting.Comment: 14 page

Employment Status and Commitment to Work in Professions

The main question in the article is to what degree underemployed and overemployed professionals are committed to the profession and the organization which employs them. Ten professional groups are included in the study and more than one in four respondents are either underemployed or overemployed. The results show that overemployment affects commitment to the profession and the organization negatively. Unexpectedly, underemployed professionals do not report weaker commitment either to the profession or to the organization compared to colleagues who work an equal but preferred number of hours