We prove that if H and δ are the Hausdorff metric and the radial metric on the space S n of star bodies in R n , with 0 in the kernel and with radial function positive and continuous, then a family A ⊂ S n that is meager with respect to H need not be meager with respect to δ. Further, we show that both the family of fractal star bodies and its complement are dense in S n with respect to δ.
Abstract. In 1998 A. Soranzo introduced the notions of +∞-and −∞-chord functions (see [16]). In this paper we give an answer to the question when a convex body is determined by the values of −∞-chord functions at chosen internal points. We also give some partial results regarding +∞-chord functions.
In this paper we consider different ways of introducing metrics in the family of star bodies. We begin with basic properties of metrics commonly used. Then we use selectors (see Definition 4.1) to extend the radial metric (see Definition 3.2) over the class of all star bodies in n-dimensional euclidean space. This way we obtain two slightly different metrics δϕ and δ ϕker (see Definition 4.2). We show that both metric spaces obtained are separable. Further, we present an analytic approach to the radial function. It enables us to define the metric δ L ϕ (see Definition 5.3). We prove that if we use δ L ϕ in the family of star bodies whose kernels have non-empty interiors, then the mapping K → kerK is continuous.
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