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Separation properties involving harmonic functions. (English) Zbl 0960.31001
The author recalls the following result from the theory of harmonic functions: let $$X$$ be a compact, convex set with non-empty interior in the space $$\mathbb{R}^m$$. Given a continuous, concave function $$u:X\to \mathbb{R}$$ and a continuous convex function $$v:X\to\mathbb{R}$$ with the property $$u\leq v$$, there exists an affine function $$a$$ in $$\mathbb{R}^m$$ with the separation property $$u\leq a\leq v$$ in $$X$$. It is also known that $$u= \inf \{a|_x$$; $$a$$, affine in $$\mathbb{R}^m$$; $$a\geq u$$ on $$X\}$$; an analogous assertion is valid for $$v$$.
The author poses a similar problem in another context. He replaces $$X$$ with $$\overline{U}$$, $$U\subset \mathbb{R}^m$$ a bounded, open set, and affine functions with harmonic ones. He replaces, also, convex and concave functions with subharmonic and superharmonic functions, respectively. Introducing the space of functions $$H(U)= \{h\in C(\overline{U})$$; $$h|_U$$ harmonic}, he proves that for superharmonic and subharmonic functions, a Hahn-Banach property does not hold with elements from $$H(U)$$ if $$m\geq 2$$. This result appears as the outcome of a long row of papers elaborated by the author since 1971.
##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 26B25 Convexity of real functions of several variables, generalizations