The caratheodory number for thek-core

I. Bárány; M. Perles

doi:10.1007/bf02123009

On Homogeneous Convex Cones, The Carathéodory Number, and the Duality Mapping

Mathematics of Operations Research ◽

10.1287/moor.26.2.234.10553 ◽

2001 ◽

Vol 26 (2) ◽

pp. 234-247 ◽

Cited By ~ 8

Author(s):

Levent Tunçel ◽

Song Xu

Keyword(s):

Convex Cones ◽

Duality Mapping ◽

Carathéodory Number ◽

Homogeneous Convex

On the Carathéodory number of interval and graph convexities

Theoretical Computer Science ◽

10.1016/j.tcs.2013.09.004 ◽

2013 ◽

Vol 510 ◽

pp. 127-135 ◽

Cited By ~ 9

Author(s):

Mitre C. Dourado ◽

Dieter Rautenbach ◽

Vinícius Fernandes dos Santos ◽

Philipp M. Schäfer ◽

Jayme L. Szwarcfiter

Keyword(s):

Carathéodory Number

Nonnegative Polynomials and Their Carathéodory Number

Discrete & Computational Geometry ◽

10.1007/s00454-014-9588-3 ◽

2014 ◽

Vol 51 (3) ◽

pp. 559-568 ◽

Cited By ~ 1

Author(s):

Simone Naldi

Keyword(s):

Nonnegative Polynomials ◽

Carathéodory Number

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Mathematische Annalen ◽

10.1007/s00208-021-02166-x ◽

2021 ◽

Author(s):

Philipp J. di Dio ◽

Mario Kummer

Keyword(s):

Moment Problem ◽

Hankel Matrix ◽

Upper Bounds ◽

Moment Problems ◽

Algebraic Varieties ◽

Lower And Upper Bounds ◽

Worst Case ◽

Flat Extension ◽

Truncated Moment Problem ◽

Carathéodory Number

AbstractIn this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

On the Carathéodory Number for Strong Convexity

Discrete & Computational Geometry ◽

10.1007/s00454-019-00169-9 ◽

2020 ◽

Author(s):

Vuong Bui ◽

Roman Karasev

Keyword(s):

Strong Convexity ◽

Carathéodory Number

On the Carathéodory Number for the Convexity of Paths of Order Three

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2011.09.018 ◽

2011 ◽

Vol 38 ◽

pp. 105-110 ◽

Cited By ~ 1

Author(s):

Rommel M. Barbosa ◽

Erika M.M. Coelho ◽

Mitre C. Dourado ◽

Dieter Rautenbach ◽

Jayme L. Szwarcfiter

Keyword(s):

Carathéodory Number

Some Steiner concepts on lexicographic products of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500608 ◽

2014 ◽

Vol 06 (04) ◽

pp. 1450060 ◽

Cited By ~ 3

Author(s):

Bijo S. Anand ◽

Manoj Changat ◽

Iztok Peterin ◽

Prasanth G. Narasimha-Shenoi

Keyword(s):

Steiner Tree ◽

Steiner Trees ◽

Partial Answer ◽

Lexicographic Product ◽

Complete Answer ◽

Radon Number ◽

Minimum Number ◽

Carathéodory Number ◽

Product Of Graphs

Let G be a graph and W a subset of V(G). A subtree with the minimum number of edges that contains all vertices of W is a Steiner tree for W. The number of edges of such a tree is the Steiner distance of W and union of all vertices belonging to Steiner trees for W form a Steiner interval. We describe both of these for the lexicographic product of graphs. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number.

The Carathéodory number of the P 3 convexity of chordal graphs

The Seventh European Conference on Combinatorics, Graph Theory and Applications ◽

10.1007/978-88-7642-475-5_34 ◽

2013 ◽

pp. 209-214

Author(s):

Erika M. M. Coelho ◽

Mitre C. Dourado ◽

Dieter Rautenbach ◽

Jayme L. Szwarcfiter

Keyword(s):

Chordal Graphs ◽

Carathéodory Number

Notes About the Carathéodory Number

Discrete & Computational Geometry ◽

10.1007/s00454-012-9439-z ◽

2012 ◽

Vol 48 (3) ◽

pp. 783-792 ◽

Cited By ~ 11

Author(s):

Imre Bárány ◽

Roman Karasev

Keyword(s):

Carathéodory Number

The Geodesic Carathéodory Number

10.5753/etc.2016.9848 ◽

2018 ◽

Author(s):

Eduardo S. Lira ◽

Diane Castonguay ◽

Erika M. M. Coelho ◽

Hebert Coelho

Keyword(s):

Carathéodory Number

Do teorema de Carathéodory surge a definição do número de Carathéodory para grafos. Este número é bem conhecido nas convexidades monofônica e de caminho de triângulos. Ele é limitado para algumas classes de grafos nas convexidades P3 e geodésica, mas apenas na convexidade P3 sabe-se que ele é ilimitado. Neste artigo, nós provamos que o número de Carathéodory é ilimitado na convexidade geodésica.