*Accepted Paper*

**Inserted:** 22 feb 2008

**Journal:** Transactions of the American Mathematical Society

**Year:** 2007

**Notes:**

http:/www.ams.org*cgi-bin*mstrack*accepted _{papers}*tran

**Abstract:**

We consider the problem of maximizing the Lebesgue measure of the convex hull of a connected compact set of prescribed one-dimensional Hausdorff measure. In dimension two, we prove that the only solutions are semicircles. In higher dimension, we prove some isoperimetric inequalities for the convex hull of connected sets, we focus on a classical open problem and we discuss a new possible approach.