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 $A (1000)$ 。