4. Let $P, P^{\prime}$ be two intersecting convex quadrilateral regions on a plane, and $O$ be a point in their intersection.

Assume that for any line passing through $O$, the segment it intercepts in region $P$ is longer than the segment it intercepts in region $P^{\prime}$. Question: Is it possible for the area ratio of region $P^{\prime}$ to region $P$ to be greater than 1.9?
4. Possible.

For any $\varepsilon>0$, construct regions $P^{\prime}$ and $P$ such that the ratio of their areas is greater than $2-\varepsilon$.

Let $O$ be the center of the square $A B C D$, and let $A^{\prime} 、 B^{\prime} 、 C^{\prime}$ be the reflections of $O$ about $A 、 B 、 C$ respectively.

Notice that $l$ is any line passing through point $O$ except for the line $A C$.

Then, the length of the segment cut by $l$ from quadrilateral $A B C D$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$ is equal.
Take points $M 、 N$ on $B^{\prime} A^{\prime} 、 B^{\prime} C^{\prime}$ respectively, such that
$$
\frac{B^{\prime} M}{B^{\prime} A^{\prime}}=\frac{B^{\prime} N}{B^{\prime} C^{\prime}}=\sqrt{1-\frac{\varepsilon}{4}} \text {. }
$$

Region $P^{\prime}$ is the convex quadrilateral $B^{\prime} M O N$ scaled from $O$ with a scaling factor of $\sqrt[4]{1-\frac{\varepsilon}{4}}$. Then this region satisfies the area ratio with region $P$ being greater than $2-\frac{\varepsilon}{2}$.