### On shape and fundamental deformation retracts II

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CONTENTSIntroduction..............................................................................................................51. Preliminaries........................................................................................................61.0. Measurable, probabilistic, and statistical spaces..............................................61.1. Transition functions..........................................................................................61.2. Linear space of P-bounded...

In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for ${L}_{p}$ metrics for all p ≥ 2 and the symmetric difference metric.

We prove that if ${\varrho}_{H}$ and δ are the Hausdorff metric and the radial metric on the space ⁿ of star bodies in ℝ, with 0 in the kernel and with radial function positive and continuous, then a family ⊂ ⁿ that is meager with respect to ${\varrho}_{H}$ 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 ⁿ with respect to δ.

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