4. Let's first consider four points $A, B, C, D$ such that one of them is inside the triangle formed by the other three (without loss of generality, let $D \in \operatorname{int} \triangle A B C$). Since among the angles $\measuredangle A D B, \measuredangle A D C, \measuredangle B D C$ there is at least one not less than $120^{\circ}$, in this case it follows that $f(A, B, C, D) \geqslant 120^{\circ}$.

Now let's consider the case when none of the points $A, B, C, D$ lies inside the triangle formed by the other three, i.e., when they form a convex quadrilateral (without loss of generality, the quadrilateral $A B C D$). Since $\measuredangle A B C + \measuredangle B C D + \measuredangle C D A + \measuredangle D A B = 360^{\circ}$, at least one of these four angles must be not less than $90^{\circ}$, so in this case it follows that $f(A, B, C, D) \geqslant 90^{\circ}$.

Therefore, we conclude $\min f(A, B, C, D) \geqslant 90^{\circ}$. If the points $A, B, C, D$ form the vertices of a rectangle, then this bound is achieved, so the solution to the problem is $90^{\circ}$.
4. For a quadruple of points $A, B, C, D$ in the plane, no three of which are collinear, let $f(A, B, C, D)$ denote the measure of the largest angle formed by these points (out of a total of 12 such angles). Determine $\min f(A, B, C, D)$, where the minimum is taken over all such quadruples of points.