The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

In 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.

The Geodesic 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.

Some Steiner concepts on lexicographic products 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.