Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 155-163.

On Mappings Preserving a Family of Star Bodies Grzegorz Sójka Wydział Matematyki i nauk Informacyjnych, Politechnika Warszawska Plac Politechniki 1, 00-668 Warszawa, Polska e-mail: grzegorz.sojka@prioris.mini.pw.edu.pl Abstract. The paper concerns the star mappings understood as topological em- edding of ℝ^{n} into itself preserving the class of bodies which are star shaped at oint 0. The main result is full characterization of star mappings (Theorem 2.8). t the end we give a solution of some related problem. SC 2000: 52A30, 54C99, 54C05 eywords: star set, star body, star mapping 1. Introduction This paper consists of two different parts, both related to [1]. Moszyńska in [1] defined a set GS (n) of transformations called “generalized star mappings”. They are positively homogeneous homeomorphisms of ℝ^{n} onto itself. That class of mappings is suitable for the notion of quotient star body (comp. [1], Prop. 2.6), however (in contrary to the statement in [1], p. 47) it is not the largest possible family of maps preserving the class of the largest family Ω^{n} of maps preserving �^{n}. In Section 3 we give a solution of Problem 1 in [1].

We use the following terminology and notation: By ℝ_{+} we denote the set �r � ℝ; r � 0� , by Σ the set of topological embeddings of ℝ_{+} into ℝ_{+} preserving 0. For affine independent points x_{1}, …, x_{n} in ℝ^{n} the simplex with vertices x_{1}, …, x_{n} is denoted by Δ (x_{1}, …, x_{n}). As usually, 𝔹^{n} and S^{n−1} are the unit ball and the unit sphere. Let A be a nonempty compact subset of ℝ^{n}; then A is a body if and only if A = cl( int (A)); the set A is called star shaped

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag