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Extremum properties of lattice packing and covering with circles. (English) Zbl 1329.05061
Let $$L$$ be a lattice in the plane and write $$\delta(L)$$ and $$\nu(L)$$ for the packing and covering density of $$L$$, respectively. Further, let $$\delta^*(L)$$ and $$\nu^*(L)$$ be the packing and covering densities of the polar lattice $$L^*$$. The paper is devoted to a uniform study of local maximum properties of functions $$\delta$$, $$\delta \delta^*$$, $$\delta \nu$$ and local minimum properties of functions $$\nu$$, $$\nu \nu^*$$, $$\nu/\delta$$. Due to the fact that the polar of a planar lattice is essentially a rotated copy of it, the results for $$\delta \delta^*$$ and $$\nu \nu^*$$ follow from those for $$\delta$$ and $$\nu$$.
The paper contains a variety of results in this direction. To describe their general flavor, we need some notation. Let $$0$$ and $$I$$ be the $$2 \times 2$$ zero and identity matrices, respectively, and let $$A = (a_{ik})$$ be a $$2 \times 2$$ symmetric matrix with trace equal to zero. Let $\| A\| = \left( \sum_{i=1}^2 \sum_{k=1}^2 a_{ik}^2 \right)^{1/2}$ be the $$\ell_2$$-norm of $$A$$. The local behavior of the density function $$\delta$$ can be analyzed by looking at the quotient $$\delta((I+A)L)/\delta(L)$$ as $$A \to 0$$. The author distinguishes four situations:
{$$\bullet$$}
if the quotient is $$\leq 1 + o(\| A\|)$$, $$\delta$$ is upper semi-stationary at $$L$$;
{$$\bullet$$}
if the quotient is $$= 1 + o(\| A\|)$$, $$\delta$$ is stationary at $$L$$;
{$$\bullet$$}
if the quotient is $$\leq 1$$, $$\delta$$ is maximum at $$L$$;
{$$\bullet$$}
if the quotient is $$\leq 1 - \mathrm{const }\| A\|$$, $$\delta$$ is ultra-maximum at $$L$$.
Here we say that the inequality holds as $$A \to 0$$ if it holds for all $$A$$ for which $$\| A\|$$ is sufficiently small and $$\mathrm{const}$$ is a positive constant. The local notions of upper semi-stationarity, stationarity, extremality and ultra-extremality can also be analogously defined for the other density functions and products under consideration. The author reviews a variety of known results (often with new proofs), as well as obtains some new ones, on lattices at which these properties are assumed; for example, not surprisingly $$\delta$$ is upper semi-stationary if and only if $$L$$ is the square or the hexagonal lattice. The author remarks that the strong property of ultra-extremality seems to be new, and proves that (more surprisingly) the packing density maximizers are also ultra-maximizers. Further, he shows that there are some rather unsymmetric parallelogram lattices for which $$\delta \nu$$ is upper semi-stationary. All in all, the paper is full of classical and new observations, confirming the special status of the hexagonal lattice, but also uncovering some unexpected properties of less remarkable lattices.
##### MSC:
 05B40 Combinatorial aspects of packing and covering 11H06 Lattices and convex bodies (number-theoretic aspects) 11H31 Lattice packing and covering (number-theoretic aspects) 11H50 Minima of forms 52C15 Packing and covering in $$2$$ dimensions (aspects of discrete geometry) 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry)
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