Set of Moskovitz dines

The Moskovitz-Dines theorem is a mathematical theorem that deals with the question of characterizing convex subsets of topological vector spaces . It comes from a work by the two mathematicians David Moskovitz and Lloyd Lyne Dines from 1939 and is closely related to two other theorems that go back to Stanisław Mazur and Errett Bishop and Robert Ralph Phelps .

Formulation of the sentence

Following the monograph by Jürg T. Marti , the sentence can be formulated as follows:

Given a topological - vector space and a contained therein closed subset comprising at least one interior point should have. ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle T\subseteq X}$

Furthermore, the condition that the regular points of form a dense subset of the boundary point set is sufficient . ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle \partial {T}}$

Then:

${\displaystyle T}$is a convex subset of . ${\displaystyle X}$

Related sentences

The theorem of Moskovitz-Dines is (for separable Banach spaces ) in a certain sense the reverse of a theorem by Stanisław Mazur from 1933, which (following Marti) can be represented as follows:

A separable -Banach space and a closed convex subset therein , which should have at least one internal point, are given. ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle T\subseteq X}$

Then the set of regular points of is a dense subset of the boundary point set . ${\displaystyle T}$ ${\displaystyle \partial {T}}$

This gives the following corollary:

If there is a separable Banach space over and a closed neighborhood of the zero point contained therein , then is a convex subset of if and only if the relation holds. ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle U\subseteq X}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle {\overline {{\partial }_{r}{T}}}=\partial {T}}$

In this context, a sentence by Bishop and Phelps ( English Bishop-Phelps support point theorem ) from 1961 is worth mentioning, which (at least in the case of the Banach spaces) elaborates the meaning of the support points in connection with convex sets:

If a closed convex subset of a -Banach space , then the set of support points of a dense subset of the boundary point set . ${\displaystyle T}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle \partial {T}}$

Explanations and Notes

• A support point is a regular point of if and only if each of its associated support functionals occurs only as a positive multiple of each of the other associated support functions. ${\displaystyle p\in \partial {T}}$ ${\displaystyle T}$
• The set of regular points of is therefore a subset of the edge of and is denoted by. ${\displaystyle T}$ ${\displaystyle \partial {T}}$ ${\displaystyle T}$ ${\displaystyle {\partial }_{r}{T}}$
• Moskovitz and Dines originally only proved their theorem for real Hilbert dreams. As Marti explains, however, the proof can be extended to any topological vector spaces without major modifications . ${\displaystyle \mathbb {R} }$
• The above theorem by Bishop and Phelps is related, albeit not identical, to the result which is known in the English-language specialist literature as Bishop-Phelps theorem or as Bishop-Phelps subreflexivity theorem and consequently every Banach space is a subreflexive space . The concept of the sub-reflective space goes back to Phelps and represents a weakening of the concept of the reflective space . A normalized space is referred to as sub- reflective if in its dual space the set of those linear functionals , which take their operator norm at a point of the unit sphere , there form a dense subset. ${\displaystyle X}$ ${\displaystyle X^{'}}$ ${\displaystyle x'\in X^{'}}$ ${\displaystyle \|x'\|}$ ${\displaystyle B_{X}\subset X}$

literature

• Errett Bishop, Robert R. Phelps: The support functionals of a convex set. In: Victor L. Klee (Ed.): Convexity. Proceedings of the seventh Symposium in Pure Mathematics of the American Mathematical Society, held at the University of Washington, Seattle, Washington June 13-15, 1961 (= Proceedings of Symposia in Pure Mathematics. 7). American Mathematical Society, Providence RI 1963, pp. 27-35, ( MR0151352 ).
• Jürg T. Marti: Convex Analysis (=  textbooks and monographs from the field of exact sciences, mathematical series . Volume 54 ). Birkhäuser, Basel / Stuttgart 1977, ISBN 3-7643-0839-7 ( MR0511737 ).
• Stanisław Mazur : About convex sets in linear normalized spaces . In: Studia Mathematica . tape 4 , 1933, pp. 70-84 , doi : 10.4064 / sm-4-1-70-84 .
• Robert E. Megginson : An Introduction to Banach Space Theory (  Graduate Texts in Mathematics . Volume 183 ). Springer, New York NY et al. 1998, ISBN 0-387-98431-3 ( MR1650235 ).
• David Moskovitz, Lloyd L. Dines: Convexity in a linear space with an inner product . In: Duke Mathematical Journal . tape 5 , 1939, pp. 520-534 ( MR0000349 ).
• Albrecht Pietsch : History of Banach Spaces and Linear Operators . Birkhäuser, Boston / Basel / Berlin 2007, ISBN 0-8176-4367-2 ( MR2300779 ).

Individual evidence

1. ^ Marti: Convex Analysis. 1977, pp. 158-161.
2. ^ Marti: Convex Analysis. 1977, p. 159.
3. ^ Marti: Convex Analysis. 1977, p. 112, p. 160.
4. ^ Marti: Convex Analysis. 1977, p. 160.
5. ^ Megginson: An Introduction to Banach Space Theory. 1998, p. 275.
6. ↑ Megginson in An Introduction to Banach Space Theory (p. 275) mentions the year 1963, in which Volume VII of the Proceedings of Symposia in Pure Mathematics appeared. The conference itself took place in 1961.
7. ^ Marti: Convex Analysis. 1977, p. 70.
8. ^ ^{A b} Marti: Convex Analysis. 1977, p. 66, p. 108.
9. ^ Marti: Convex Analysis. 1977, p. 158.
10. ^ Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 81.
11. ^ Megginson: An Introduction to Banach Space Theory. 1998, pp. 270-279.