  To determine the sharp constants for the one dimensional Lieb--Thirring
inequalities with exponent gamma in (1/2,3/2) is still an open problem.
According to a conjecture by Lieb and Thirring the sharp constant for these
exponents should be attained by potentials having only one bound state. Here we
exhibit a connection between the Lieb--Thirring conjecture for gamma=1 and an
isporimetric inequality for ovals in the plane.
