Originally Posted by
undefined I haven't studied this formally per se, but both (1) and (2) seem to fail when we consider the simple case of two intersecting spheres in R^3, or even just intersecting discs in R^2.
Edit: Actually easier to write in $\displaystyle \,\mathbb{R}$,
let $\displaystyle A = [0,3], C = [1,2]$,
so
$\displaystyle B(A) = \{0, 3\}$,
$\displaystyle B(C) = \{1, 2\}$,
$\displaystyle B(A \cup C) = \{0, 3\}$,
$\displaystyle B(A) \cup B(C) = \{0,1,2,3\}$,
$\displaystyle B(A \cap C) = \{1,2\}$,
$\displaystyle B(A) \cap B(C) = \emptyset$