Abstract. A family of sets ${\cal F} = \{C_\alpha\} $ in Euclidean space ${\msbm E}^d$ has the {\it strong binary intersection} (SBI) property provided for any selection $\{v_\alpha\} $ of vectors in ${\msbm E}^d$ the family of translates ${\cal F}' = \{v_\alpha + C_\alpha\mid C_\alpha\in {\cal F}\} $ has nonempty intersection if and only if any two members of ${\cal F}'$ have nonempty intersection. We show that a finite family of at least five convex bodies in ${\msbm E}^d,$ $d=2,3$, has the SBI property if and only if the members of ${\cal F}$ are isothetic parallelotopes, i.e., the edges of these parallelotopes are parallel to some $d$ linearly independent vectors in ${\msbm E}^d$.

AMS Subject Classification (1991): 52A35

Received April 15, 2005, and in revised form November 17, 2005. (Registered under 5/2005.)