  A well-known result says that the Euclidean unit ball is the unique fixed
point of the polarity operator. This result implies that if, in $\mathbb{R}^n$,
the unit ball of some norm is equal to the unit ball of the dual norm, then the
norm must be Euclidean. Motivated by these results and by relatively recent
results in convex analysis and convex geometry regarding various properties of
order reversing operators, we consider, in a real Hilbert space setting, a more
general fixed point equation in which the polarity operator is composed with a
continuous invertible linear operator. We show that if the linear operator is
positive definite, then the considered equation is uniquely solvable by an
ellipsoid. Otherwise, the equation can have several (possibly infinitely many)
solutions or no solution at all. Our analysis yields a few by-products of
possible independent interest, among them results related to coercive bilinear
forms (essentially a quantitative convex analytic converse to the celebrated
Lax-Milgram theorem from partial differential equations) and a characterization
of real Hilbertian spaces.
