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Various categorical approaches to statistical spaces

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

Optimal isometries for a pair of compact convex subsets of ℝⁿ

Irmina HerburtMaria Moszyńska — 2009

Banach Center Publications

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.

Concerning Sets of the First Baire Category with Respect to Different Metrics

Maria MoszyńskaGrzegorz Sójka — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that if ϱ 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 ϱ 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.

Fractal star bodies

Irmina HerburtMaria MoszyńskaDorette Pronk — 2009

Banach Center Publications

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.

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