  One isoperimetric inequality for the fundamental sloshing eigenvalue is
derived under the assumption that containers have vertical side walls and
either finite or infinite depth. It asserts that among all such containers,
whose free surfaces are convex, have two axes of symmetry and a given perimeter
length, this eigenvalue is maximized by infinitely deep ones provided the free
surface is either the square or the equilateral triangle. The proof is based on
the recent isoperimetric result obtained by A. Henrot, A. Lemenant and
I.~Lucardesi for the first nonzero eigenvalue of the two-dimensional Neumann
Laplacian under the perimeter constraint.
  Another isoperimetric inequality for the fundamental eigenvalue, which
describes sloshing in containers with vertical walls, is a consequence of the
classical result due to G. Szeg\H o concerning the first nonzero eigenvalue of
the free membrane problem.
