Four, find the minimum perimeter of a convex pentagon on a Cartesian plane whose vertices are lattice points (i.e., the x and y coordinates of each vertex are integers).

 untranslated text:
求平面直角坐标系中格点凸五边形 (即每个顶点的纵、横坐标都是整数的凸五边形)的周长的最小值.

 translated text:
Find the minimum perimeter of a convex pentagon on a Cartesian plane whose vertices are lattice points (i.e., the x and y coordinates of each vertex are integers).
Let the 5 vertices of this convex pentagon be $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$, with coordinates $A_{j}(x_{j}, y_{j})$, and represent vertex $A_{j}$ as $x_{j} + i y_{j}, j=1,2,3,4,5, i$ being the imaginary unit.
Let $z_{j} = A_{j+1} - A_{j}, j=1,2,3,4,5, A_{6} = A_{1}$, then
(1) The real and imaginary parts of $z_{j}$ are integers, $i|i<j| \neq 0$ (thus $\left|z_{j}\right| \geqslant 1$);
(2) $x_{1} + z_{2} + z_{3} + z_{1} + z_{6} = 0$;
(3) The perimeter of the convex pentagon $A_{1} A_{2} \cdots A_{5}$ is $\left|z_{1}\right| + \left|z_{2}\right| + \cdots + \left|z_{5}\right|$.

By convexity, any two $z_{j}, z_{k} (j \neq k)$ do not have the same direction. From (1), if some $z_{j}$ satisfies $\left|z_{j}\right| \neq 1$, then $\left|z_{j}\right|$ can only be $\sqrt{2}, 2, \sqrt{5}, \cdots$.
The number of $z_{1}, z_{2}, \cdots, z_{5}$ with modulus 1 is at most 4: $\pm 1, \pm i$.
(1) If $\left|z_{1}\right|, \left|z_{2}\right|, \cdots, \left|z_{5}\right|$ have exactly 4 ones, by (2), the remaining one is 0, which contradicts (1).
(2) If $\left|z_{1}\right|, \left|z_{2}\right|, \cdots, \left|z_{3}\right|$ have exactly 3 ones, the remaining two are $\sqrt{2}$ (the modulus of $\sqrt{2}$ is at most 4: $1 \pm i$, $-1 \pm i$), then they do not satisfy (2). Thus, the perimeter is not less than $1 + 1 + 1 + \sqrt{2} + 2 = 5 + \sqrt{2}$.
(3) If $\left|z_{1}\right|, \left|z_{2}\right|, \cdots, \left|z_{5}\right|$ have exactly 2 ones, the remaining 3 are $\sqrt{2}$, as shown in the figure, the perimeter is $1 + 1 + \sqrt{2} + \sqrt{2} + \sqrt{2} = 2 + 3 \sqrt{2}$.
(4) In other cases, the perimeter is not less than $1 + 4 \sqrt{2}$.

In summary, the minimum perimeter of a lattice convex pentagon is $2 +$
$$
3 \sqrt{2} \text{. }
$$