We give a light talk on optimality of shapes in geometry and physics.
First, we recollect classical geometric results that the disk
has the largest area (respectively, the smallest perimeter)
among all domains of a given perimeter (respectively, area).
Second, we recall that the circular drum has the lowest
fundamental tone among all drums of a given area or perimeter
and reinterpret the result in a quantum-mechanical language of nanostructures.
In parallel, we discuss the analogous optimality of square
among all rectangles in geometry and physics.
As the main body of the talk,
we present a joint work with Freitas in which we show that
the disk actually stops to be the optimiser for elastically supported membranes,
disproving in this way a long-standing conjecture of Bareket's.
We also present our recent attempts to prove the same spectral-geometric properties
in relativistic quantum mechanics.
It is frustrating that such an illusively simple
and expected result remains unproved
and apparently out of the reach of current mathematical tools.
