A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?
1. **Identify the problem**: We need to find the point \( P \) inside a convex rectangular garden \( ABCD \) such that the sum of the distances from \( P \) to the four corners \( A, B, C, \) and \( D \) is minimized.

2. **Use the triangle inequality**: For any point \( P \) inside the rectangle, we can use the triangle inequality to establish that:
   \[
   PA + PC \geq AC
   \]
   with equality if and only if \( P \) lies on the line segment \( \overline{AC} \). Similarly,
   \[
   PB + PD \geq BD
   \]
   with equality if and only if \( P \) lies on the line segment \( \overline{BD} \).

3. **Sum the inequalities**: Adding these two inequalities, we get:
   \[
   PA + PB + PC + PD \geq AC + BD
   \]
   with equality if and only if \( P \) is the intersection of the diagonals \( AC \) and \( BD \).

4. **Conclusion**: The point \( P \) that minimizes the sum \( PA + PB + PC + PD \) is the intersection of the diagonals \( AC \) and \( BD \). In a rectangle, the diagonals bisect each other, so the intersection point is the center of the rectangle.

\[
\boxed{\text{The point } P \text{ should be placed at the center of the rectangle.}}
\]