Containment and Circumscribing Simplices

The following containment theorem is presented: If K and L are convex bodies such that every simplex that contains L also contains some translate of K , then in fact the body L must contain a translate of the body K . One immediate consequence of this theorem is a strengthened version of Weil's mixed-volume characterization of containment. Received October 31, 1995, and in revised form February 28, 1996.