A Characterization of Extremal Sets in Hilbert Spaces
We give a characterization of extremal sets in Hilbert spaces that generalizes a classical theorem of H. W. E. Jung. We investigate also the behaviour of points near to the circumsphere of such a set with respect to the Kuratowski and Hausdorff measures of non-compactness.
N ov 2 00 4 A Note on Self-extremal Sets in L p ( Ω ) Spaces by
In [5], [6] we introduced the notion of (self-) extremal sets of a Banach space (X, ‖ · ‖). For a non-empty bounded subset A of X we denote by r(A) the relative Chebyshev radius of A with respect to the closed convex hull coA of A, i.e. r(A) := inf y∈coA sup x∈A ‖x−y‖. The self-Jung constant of X is defined by Js(X) : = sup {r(A) : A ⊂ X, with diam(A) = 1}. If in this definition we replace r(A) by the relative Chebyshev radius rX(A) of A with respect to the wholeX , we get the Jung constant J(X) ofX . Recall that a bounded subset A ofX consisting of at least two points is said to be extremal (resp. self-extremal) if rX(A) = J(X) d(A) (resp. r(A) = Js(X) d(A)).
Distribution-free monitoring of univariate processes
We consider statistical process control (SPC) of univariate processes when observed data are not normally distributed. Most existing SPC procedures are based on the normality assumption. In the literature, it has been demonstrated that their performance is unreliable in cases when they are used for monitoring non-normal processes. To overcome this limitation, we propose two SPC control charts for applications when the process data are not normal, and compare them with the traditional CUSUM chart and two recent distribution-free control charts. Some empirical guidelines are provided for practitioners to choose a proper control chart for a specific application with non-normal data.