[Paper Review] A Characterization of Extremal Sets in Hilbert Spaces

V. NguyenKhac, K. NguyenVan|ArXiv.org|Mar 19, 2002

Advanced Banach Space Theory4 references3 citations

TL;DR

This paper characterizes extremal sets in Hilbert spaces by generalizing Jung's classical theorem, proving that a bounded set $ A $ with diameter $ \sqrt{2} $ is extremal if and only if it contains $ p $-simplices with all edges arbitrarily close to $ \sqrt{2} $, and showing that such sets have Hausdorff measure of non-compactness $ \chi(A) = 1 $. The result extends the understanding of extremal configurations in infinite-dimensional Hilbert spaces and links their geometric structure to measures of non-compactness.

ABSTRACT

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.

Motivation & Objective

• To generalize Jung’s classical theorem on extremal sets in Euclidean spaces to infinite-dimensional Hilbert spaces.
• To characterize extremal sets in Hilbert spaces using the existence of dense $ p $-simplices with edges close to the diameter.
• To analyze the behavior of points near the circumsphere of extremal sets using Kuratowski and Hausdorff measures of non-compactness.
• To establish a precise relationship between the geometric structure of extremal sets and their non-compactness properties.

Proposed method

• Uses the definition of extremal sets as those achieving the maximal Chebyshev radius relative to diameter, with $ r(A) = J_s(H) d(A) = \frac{1}{\sqrt{2}} d(A) $.
• Applies the concept of Chebyshev centers and their convergence properties in reflexive Hilbert spaces to derive geometric constraints on point configurations.
• Employs the 'Mushroom Lemma' (from [3]) to analyze the contribution of points near the circumsphere to non-compactness.
• Uses contradiction and covering arguments with balls and sets of bounded diameter to bound the Kuratowski measure of non-compactness $ \alpha(A) $.
• Constructs a sequence of $ p $-simplices within $ A $ with edge lengths approaching $ \sqrt{2} $, using iterative selection of points with large mutual distances.
• Establishes the key inequality $ \sum t_i \|y_i - y_j\|^2 > 2 - \frac{4}{n} $ to derive bounds on weights and distances, leading to the existence of dense simplices.

Experimental results

Research questions

• RQ1 What characterizes extremal sets in infinite-dimensional Hilbert spaces, and how do they generalize the finite-dimensional case?
• RQ2 How does the Hausdorff measure of non-compactness $ \chi(A) $ behave for extremal sets in Hilbert spaces?
• RQ3 What is the role of points near the circumsphere in contributing to the measure of non-compactness of an extremal set?
• RQ4 Can the existence of $ p $-simplices with edges arbitrarily close to the diameter fully characterize extremal sets in Hilbert spaces?

Key findings

• A bounded set $ A \subset H $ with $ d(A) = \sqrt{2} $ is extremal if and only if for every $ \varepsilon > 0 $ and every $ p \in \mathbb{N} $, $ A $ contains a $ p $-simplex with all edge lengths at least $ \sqrt{2} - \varepsilon $.
• The Hausdorff measure of non-compactness of an extremal set $ A $ with $ r(A) = 1 $ is exactly $ \chi(A) = 1 $, indicating maximal non-compactness in this normalized setting.
• The Kuratowski measure of non-compactness $ \alpha(A) $ of such a set satisfies $ \alpha(A) = \sqrt{2} $, showing that the set cannot be covered by finitely many sets of diameter less than $ \sqrt{2} $.
• The main contribution to the measure of non-compactness comes from points near the circumsphere of the extremal set, as shown via the 'Mushroom Lemma' and covering arguments.
• The Chebyshev center of any such $ p $-simplex lies in the convex hull of its vertices on the sphere of radius $ r_n > 1 - \frac{1}{n} $, ensuring geometric concentration.
• The Chebyshev radius $ r' $ of a $ p $-simplex with edges $ \geq \sqrt{2} - \varepsilon $ satisfies $ r' \to 1 $ as $ p \to \infty $, confirming the extremality condition.