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I am the down voter. This does not appear to be a research-level question. Why can't you just take the expectation value of the area of the cell containing the i-th point? Clearly this should not depend on i.
Also, Hermite defined the Hermite constant (equivalent up to some factors to $\delta_n$) way before Mahler came around. Maybe it's worth tracking the original reference?
Not exactly elementary, but the Voronoi theory of perfect quadratic forms is another usual route to define the densest lattice packing, and does not usually invoke Mahler's selection principle. See for example: books.google.com/…
Number of partitions of $n$ into no more than $k$ terms that are each no larger than $l$. The symmetry between $l$ and $k$ might not be immediately obvious to novices.
Under the same heading: equality of the mutual inductance $M_{12}$, ratio of the emf induced in coil 1 to the rate of change of current in coil 2, to $M_{21}$.
P.S. The map will have $\rho'\ge\rho$ because a uniform random point in $P$ has a smaller distance from the origin than a uniform random point in $S$ (which has the same distribution of distances from the origin as a random point in the disk of radius $r$).
Let $C_\rho$ denote the centered circle of radius $\rho$. Given $\rho$ and $\rho'$, you could map the intersection $C_\rho\cap P$ to an arc $C_{\rho'}\cap S$ in some canonical way. Now you only have to figure out the unique monotonic map $\rho\mapsto\rho'$ that would make this area preserving when applied to all of $P$. That map only depends on the radial profiles of $P$ and $S$.
@JosephO'Rourke: pick any nice odd function $f:S^2\to\mathbf{R}$, and let $\rho=\sqrt{1+\alpha f}$. If you care about convexity, then $\alpha$ needs to be sufficiently small to preserve convexity. The fact that any section through the origin has the same area is because the area is given by $\tfrac{1}{2}\int_{S\cap S^2} \rho^2 d\theta = \tfrac{1}{2}\int_{S\cap S^2} (1+\alpha f) d\theta = \pi$.
The areas of sections of a body are given by the spherical Radon transform applied to $\rho^{n-1}$, where $\rho$ is the radial distance from the origin to the boundary. Therefore, it is easy to construct equiareal bodies by considering the kernel of the transform, which is just the odd functions. However, every equiareal centrally-symmetric body is indeed a sphere.
You should be able to solve for $f^*$ directly. Let $f(x)=g(x)^2$, then you have the following stationarity condition: $a + 2(b||x|| -\mu) g(x)=0$, where $\mu$ is a Lagrange multiplier. So you just have to solve for the $\mu$ that gives $\int_C g(x)^2=1$.
