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Rational realization of the minimum ranks of nonnegative sign pattern matrices

Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yan Ling Shao, Lihua Zhang (2016)

Czechoslovak Mathematical Journal

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set { + , - , 0 } ( { + , 0 } , respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix 𝒜 is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of 𝒜 . Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in ℝ r - 1 and Steinitz’s theorem on the rational realizability of...

Reduced spherical polygons

Marek Lassak (2015)

Colloquium Mathematicae

For every hemisphere K supporting a spherically convex body C of the d-dimensional sphere S d we consider the width of C determined by K. By the thickness Δ(C) of C we mean the minimum of the widths of C over all supporting hemispheres K of C. A spherically convex body R ⊂ S d is said to be reduced provided Δ(Z) < Δ(R) for every spherically convex body Z ⊂ R different from R. We characterize reduced spherical polygons on S². We show that every reduced spherical polygon is of thickness at most π/2. We...

Regularization of star bodies by random hyperplane cut off

V. D. Milman, A. Pajor (2003)

Studia Mathematica

We present a general result on regularization of an arbitrary convex body (and more generally a star body), which gives and extends global forms of a number of well known local facts, like the low M*-estimates, large Euclidean sections of finite volume-ratio spaces and others.

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