This paper is concerned with the algebraic Riccati equations (AREs) related to the H-infinity filtering problem. A necessary and sufficient condition for the H-infinity problem to be solvable is that the H-infinity ARE has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite. It is shown that such a stabilizing solution is a monotonically non-increasing convex function of the prescribed H-infinity norm bound gamma. This property of the H-infinity ARE is very important for the analysis of the performance of the H-infinity filter. In this paper, the size of the set of all H-infinity filters is considered on the basis of the monotonicity of the above Riccati solution. It turns out that, under a certain condition, the degree of freedom of the H-infinity filter reduces at the optimal H-infinity norm bound. These results provide a guideline for selecting the value of gamma. Some numerical examples are included.