Limit shape of optimal convex Lattice polygons in the sense of different metrics. (Q1408879)

Let \(p\in\mathbb{Z}^+\) or infinity. For \((x_1,y_1)\), \((x_2,y_2)\) two points in the plane, the \(\ell_p\)-distance between them is defined by \(\bigl(| x_2-x_1| ^p+| y_1-y_2| ^p\bigr)^{1/p}\). A convex lattice \(n\)-gon (i.e., a polygon whose \(n\) vertices are points of the integer lattice) is said to be optimal in the sense of the \(\ell_p\)-metric if it has minimal \(\ell_p\)-perimeter with respect to the number of vertices \(n\). This optimal \(n\)-agon is denoted by \(Q_p(n)\). As main result of this paper, the author determines the limit shape of a sequence of optimal convex lattice polygons \(Q_p(n)\) when \(n\) tends to infinity: the limit shape of the south-east arcs of the sequences \(Q_p(n)\) is a curve \(\gamma_p\) whose parametrization is given explicitly; the limit shapes of the other three arcs are suitable rotations and translations of the same curve \(\gamma_p\). This theorem is first proved for particular values of \(n\) (a certain sequence of \(\mathbb{N}\)), and then for the general case in two steps, depending on \(n\) is an even or an odd number.