  In 1934, Reinhardt asked for the centrally symmetric convex domain in the
plane whose best lattice packing has the lowest density. He conjectured that
the unique solution up to an affine transformation is the smoothed octagon (an
octagon rounded at corners by arcs of hyperbolas). This article offers a
detailed strategy of proof. In particular, we show that the problem is an
instance of the classical problem of Bolza in the calculus of variations. A
minimizing solution is known to exist. The boundary of every minimizer is a
differentiable curve with Lipschitz continuous derivative. If a minimizer is
piecewise analytic, then it is a smoothed polygon (a polygon rounded at corners
by arcs of hyperbolas). To complete the proof of the Reinhardt conjecture, the
assumption of piecewise analyticity must be removed, and the conclusion of
smoothed polygon must be strengthened to smoothed octagon.
