منابع مشابه
About the Carathéodory Number
In this paper we give sufficient conditions for a compactum in R to have Carathéodory number less than n+ 1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg-type theorem for families of convex compacta.
متن کاملOn the Integral Carathéodory Property
In this note we document the existence of a finitely generated rational cone that is not covered by its unimodular Hilbert subcones, but satisfies the integral Carathéodory property. We explain the algorithms that decide these properties and describe our experimental approach that led to the discovery of the examples.
متن کاملCarathéodory bounds for integer cones
Let b ∈ Zd be an integer conic combination of a finite set of integer vectors X ⊂ Zd . In this note we provide upper bounds on the size of a smallest subset X̃ ⊆ X such that b is an integer conic combination of elements of X̃ . We apply our bounds to general integer programming and to the cutting stock problem and provide an NP certificate for the latter, whose existence has not been known so far.
متن کاملNonpathological Lyapunov functions and discontinuous Carathéodory systems
Differential equations with discontinuous righthand side and solutions intended in Carathéodory sense are considered. For these equations sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven.
متن کاملComputational Aspects of the Colorful Carathéodory Theorem
Let P1, . . . , Pd+1 ⊂ R be d-dimensional point sets such that the convex hull of each Pi contains the origin. We call the sets Pi color classes, and we think of the points in Pi as having color i. A colorful choice is a set with at most one point of each color. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the com...
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ژورنال
عنوان ژورنال: Annales Academiae Scientiarum Fennicae Series A I Mathematica
سال: 1976
ISSN: 0066-1953
DOI: 10.5186/aasfm.1976.0215