Computing smallest convex intersecting polygons

A polygon C is an intersecting polygon for a set (Present Formula) of objects in R² if C intersects each object in (Present Formula), where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set (Present Formula) of objects. We present an FPTAS for both problems for the case where (Present Formula) is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minim umperimeter intersecting polygon for the case where (Present Formula) is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum-perimeter intersecting polygon of lines or of disjoint segments.