Connection between the Lieb--Thirring conjecture for Schroedinger   operators and an isoperimetric problem for ovals on the plane
  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. 
Rafael D. Benguria and Michael Loss