A continuous movement version of the Banach Tarski paradox: A solution to de Groot's problem

In 1924 Banach and Tarski demonstrated the existence of a Paradoreal decomposition of the 3-ball B. i.e.. it piecewise isometry front R onto tiki copie, 4 B Hit, article answers it question of de Groot from 1959 by showing that there I., it paradoxical decomposition of B which the pieces move continuously while remaining disjoint to yield two copies of B More generally we show that if n >= 2. any two bounded sets in R-n that are equidecomposable with proper isometers are continously equidecomposable in this sense.