The isoperimetric problem for the Favard length

The Favard length of a Borel set $E$ on the Euclidean plane is the average length of its orthogonal projections (mean shadow). Here we solve the following problem: Minimize the Favard length in the class of planar Borel sets with unit area (2-dimensional Lebesgue measure). It is shown that a circle with unit area is a solution, as expected.