@MISC{Alimov02onstrict, author = {A. R. Alimov}, title = {ON STRICT SUNS IN ℓ ∞ (3)}, year = {2002} }

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Abstract

A subset M of a normed linear space X is said to be a strict sun if, for every point x ∈ X \M, the set of its nearest points from M is non-empty and if y ∈ M is a nearest point from M to x, then y is a nearest point from M to all points from the ray {λx + (1 − λ)y | λ> 0}. In the paper there obtained a geometrical characterisation of strict suns in ℓ ∞ (3). In comparison with [1] we establish a more precise property of stict suns. A subset M of a normed linear space X is said to be a strict sun if, for every x ∈ X \ M, the set of its nearest points from M is non-empty and if y ∈ M is the nearest point to x, then y is the nearest ponit from M to every point from the ray starting at y and passing through x. In this paper we obtain a geometric characterisation of strict suns in the space ℓ ∞ (3) (Theorem 1). This characterisation provides a more precise property of stricts suns in ℓ ∞ (3) than Theorem A (see below) from [1] does. 1. Definitions and notation. We will consider only real spaces. Following K. Menger [2] and H. Berens and L. Hetzelt [3], a set M ⊂ R n is called ℓ 1-convex if, for all x, y ∈ M, x ̸ = y, there is a point z ∈ M, z ̸ = x, z ̸ = y, such that ‖x − y‖ℓ 1 = ‖x − z‖ℓ 1 + ‖z − y‖ℓ 1 (here ‖·‖ℓ 1 is the standard ℓ1-norm on R n). As it is shown in [3], ℓ 1-convexity (together with closedness) proved to be a characteristic property of suns ℓ ∞ (n). (Here we recall that a set M ⊂ X is a sun if, for every point x ∈ X \ M there is a point y, that is nearest from M to x and such that y is a nearest point to all points from the ray starting at y and passing through x; such a point y is called a solar point for x). It is clear that every strict sun is a sun. The inverse statement is not true in general. On suns, strict suns and other approximative sets see [4], [5]. An elegant characterisation of suns in ℓ ∞ (n) was obtained by H. Berens and L. Hetzelt [3] (see also [6]). Theorem (H. Berens, L. Hetzelt). A closed nonvoid set M ⊂ R n is a sun in ℓ ∞ (n) if and only if it is ℓ 1-convex. The first characterisation of strict suns in ℓ ∞ (n) in geometrical terms was independently obtained by B. Brosowski [7] and Ch. Dunham [8]. In [1] (Theorem A) there obtained a characterisation of strict suns in ℓ ∞ (n), which is similar to the metric characterisation due to Berens and Hetzelt.