Roswitha Blind (also published as Roswitha Hammer) [1] [2] is a German mathematician, specializing in convex geometry, discrete geometry, and polyhedral combinatorics, and a politician and organizer for the Social Democratic Party of Germany in Stuttgart.
As Roswitha Hammer, Blind completed a Ph.D. in 1974 at the University of Stuttgart. Her dissertation, Über konvexe Strukturen und die Beziehungen zur elementaren Konvexität, concerned convex geometry and discrete geometry and was supervised by Kurt Leichtweiss. [2]
She is best known in mathematics for a 1987 publication with Peter Mani-Levitska in which, solving a conjecture of Micha Perles, she and Mani-Levitska proved that the combinatorial structure of simple polytopes is completely determined by their graphs. [3] This result has been called the Blind–Mani theorem [4] or the Perles–Blind–Mani theorem. [5]
In a 1979 publication, [6] she introduced a class of convex polytopes sometimes called the Blind polytopes, generalizing the semiregular polytopes and Johnson solids, in which all faces are regular polytopes. [7]
Blind became a city councillor in the Möhringen-Vaihingen district of Stuttgart in 2004, [8] stepping down from that seat in 2009 in order to become chair of the Social Democratic Party of Germany local council group. [9] As councillor, in order to better serve the youth of her district, she became chair of a local football club, 1. FC Lauchhau-Lauchäcker, in 2006, also serving as president of the Stuttgart Sports Forum. [10]
She retired from politics in 2014, [8] and from her position with the football club in 2016. [10]
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician Robert P. Dilworth (1950).
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.
In mathematics, the "happy ending problem" is the following statement:
In geometry, a d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges. The vertex figure of a simple d-polytope is a (d − 1)-simplex.
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.
In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively.
In geometry and polyhedral combinatorics, a k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a k-neighborly polytope requires a dimension of 2k or more. A d-simplex is d-neighborly. A polytope is said to be neighborly, without specifying k, if it is k-neighborly for . If we exclude simplices, this is the maximum possible k: in fact, every polytope that is k-neighborly for some is a simplex.
In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form
In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956.
In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric polytope has at least 3d nonempty faces.
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by W. T. Tutte (1963), states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.
In discrete and computational geometry, a set of points in the Euclidean plane or a higher-dimensional Euclidean space is said to be in convex position or convex independent if none of the points can be represented as a convex combination of the others. A finite set of points is in convex position if all of the points are vertices of their convex hull. More generally, a family of convex sets is said to be in convex position if they are pairwise disjoint and none of them is contained in the convex hull of the others.
Peter McMullen is a British mathematician, a professor emeritus of mathematics at University College London.
A flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs.
Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.