Carathéodory's theorem and \(H\)-convexity (Q5930024)

Let \(H\) be a subset of the unit sphere \(S^{n-1}\) of the Euclidean space \({\mathbb{R}}^n\) which is not one-sided (i.e. \(H\) is not contained in any closed hemisphere of \(S^{n-1}\)). By \(\text{md} H\) we denote the largest positive integer \(k\) such that there are vectors \(a_0, a_1, \ldots ,a_k\in H\) with the following properties: (i) the vectors are positively dependent, (ii) every \(k\) of the vectors is linearly independent. This functional was introduced by the first author in 1976. NEWLINENEWLINENEWLINEIn this paper the connection of the classical Carathéodory Theorem and the \(md\) functional is investigated. NEWLINENEWLINENEWLINELet \(H\) be a subset of the unit sphere \(S^{n-1}\) which is not one-sided. The Carathéodory number \(\text{ctd}(H)\) of the set \(H\) is the minimal integer \(k\) such that for every compact set \(K\subset {\mathbb{R}}^n\) the equality NEWLINE\[NEWLINEcl\Bigl( \bigcup _{a_0, a_1,\ldots , a_k\in K} \text{conv}_H \{a_0, a_1,\ldots , a_k\}\Bigr) =\text{conv}_H KNEWLINE\]NEWLINE holds, where \(cl\) denotes closure. Moreover, if the equality does not hold for any integer \(k\), then we assume that \(\text{ctd}(H)=\infty\). NEWLINENEWLINENEWLINEIn the article the following statements are proved. For every subset \(H\) of \(S^{n-1}\) that is not one-sided, the inequality \(\text{ ctd} (H)\geq \text{md}(H)\) holds. For every subset \(H\subset S^{n-1}\) that is not one-sided, the equality \(\text{ctd}(H)= \text{ctd} (cl H)\) holds. In particular, if \(cl H=S^{n-1}\), then \(\text{ctd}(H)=n\). NEWLINENEWLINENEWLINEThere are more statements that provide details about the connection between \(md\) and \(ctd\) by giving estimates of \(\text{ctd}(H)\) for given values of \(n\) and \(\text{md}(H)\).