Problem 742

Minimum area of a convex grid polygon

A symmetrical convex grid polygon is a polygon such that:

All its vertices have integer coordinates.
All its internal angles are strictly smaller than $180^\circ$.
It has both horizontal and vertical symmetry.

For example, the left polygon is a convex grid polygon which has neither horizontal nor vertical symmetry, while the right one is a valid symmetrical convex grid polygon with six vertices:

Define $A(N)$, the minimum area of a symmetrical convex grid polygon with $N$ vertices.

You are given $A(4) = 1$, $A(8) = 7$, $A(40) = 1039$ and $A(100) = 17473$.

Find $A(1000)$.

格点凸多边形的最小面积

对称格点凸多边形是指满足以下条件的多边形：

所有顶点的坐标均为整数。
所有内角严格小于$180^\circ$。
在水平和竖直方向上对称。

例如，下图左的多边形是格点凸多边形，但是在水平和竖直方向上均不对称；而下图右的多边形则是一个有六个顶点的对称格点凸多边形：

记$A(N)$为所有有$N$个顶点的对称格点凸多边形的最小面积。

已知$A(4) = 1$，$A(8) = 7$，$A(40) = 1039$，$A(100) = 17473$。

求$A(1000)$。