Convergence in Hausdorff metric preserves geometric shape. (English) Zbl 0942.68113

Radunović, B.

Mat. Vesn. 48, No. 1-2, 25-28 (1996).

Let us denote by \(G_n\) a group of transformations of \(R^n\) generated by homoteties, translations and rotations. Two compact subsets \(F\) and \(F'\) of \(R^n\) are said to have the same geometric shape if there exists a transformation \(h\in G_n\) such that \(F'=h[F].\)
This paper deals with the interesting question of invariance of geometric shape with respect to the Hausdorff convergence. The author resolves this question completely by proving that the limit of a convergent sequence of compact sets in \(R^n\) of the same geometric shape is again a set of the same geometric shape or a singleton.

Reviewer: Siniša Vrećica (Beograd)

MSC:

68T10	Pattern recognition, speech recognition
54B20	Hyperspaces in general topology

Keywords:

geometric shape; Hausdorff metric; general position