Andromaque and extrema

I’ve just read Racine’s Andromaque – I intend to see it soon at the Comédie Française, and I thought I should prepare. I am curious. Upon reaching the last lines of Hermione while reading the text alone, could you suppress the urge to say bitch?

Several elementary optimisation problems have highly symmetrical solutions. For example, among triangles of given perimeter, the equilateral triangle has maximal area. Even more strikingly, among all curves of given perimeter, the circle encloses maximal area (Dido’s problem).

Is there any reason for this? My dad actually asked me recently, and the best I could come up with was – well, if an extremum is not symmetric, it must have high multiplicity (as all its reflections and rotations must also be extrema), and then… Well, I can’t really see how to give a general argument that leads to a contradiction from here. Steiner’s work (see Polya’s book for a good exposition) goes in the right direction, but I do not know whether that technique is of general applicability. How would you go about things?

It would also be nice to have a few examples of elementary problems about maxima and minima that have symmetric statements but non-symmetric solutions. I am vaguely aware of the fact that there are situations in physics where symmetric forces give rise to non-symmetric stable configurations – surely that’s related?


About valuevar

I am a number theorist with side interests in combinatorics and group theory.
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