Introduction:
In this problem, we are given a quadrilateral ABCD and we need to determine whether AB + BC + CD + DA is greater than AC + BD or not. To solve this problem, we will use the triangle inequality theorem and properties of quadrilaterals.

Triangle Inequality Theorem:
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem can be extended to quadrilaterals as well.

Properties of Quadrilaterals:
1. In a quadrilateral, the sum of the lengths of any two sides is always greater than the lengths of the other two sides.
2. The sum of the lengths of opposite sides in a quadrilateral is always equal.

Analysis:
In quadrilateral ABCD, let's consider the lengths of its sides:
- AB = a
- BC = b
- CD = c
- DA = d

We need to determine whether AB + BC + CD + DA is greater than AC + BD or not.

Case 1: AC + BD = a + c + b + d
In this case, AB and CD are opposite sides of the quadrilateral and BC and DA are the other pair of opposite sides.
According to the properties of quadrilaterals, the sum of the lengths of opposite sides is always equal. Therefore, AB + CD = BC + DA.
So, AB + BC + CD + DA = a + b + c + d.

Case 2: AC + BD > a + c + b + d
In this case, AC + BD is greater than the sum of all sides of the quadrilateral. This is not possible as the sum of the lengths of any two sides of a quadrilateral is always greater than the lengths of the other two sides. Therefore, this case is not valid.

Conclusion:
From the above analysis, we can conclude that AB + BC + CD + DA is always equal to a + b + c + d, which is equal to AC + BD. Therefore, AB + BC + CD + DA is not greater than AC + BD.