probable squares in a square cake There is a probability density function defined on the square [0,1]x[0,1].
The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area.
Now Alice and Bob play a game: Alice marks two disjoint squares, Bob chooses the square that contains the maximum probability, and Alice gets the other square. The goal of Alice is to maximize the probability in her square.
Obviously, in some cases Alice can assure herself a probability of 1/2, for example, if the pdf is uniform in [0,$1 \over 2$]x[0,1], she can cut the squares [0,$1 \over 2$]x[0,$1 \over 2$] and [0,$1 \over 2$]x[$1 \over 2$,1], both of which contain $1 \over 2$.
However, in other cases Alice can assure herself only $1 \over 4$, for example, if the pdf is uniform in [0,1]x[0,1].
Are there pdfs for which Alice cannot assure herself even $1 \over 4$ ?
What is the worst case for Alice?

I assume by disjoint squares you mean squares made with 2 cuts, each parallel to an axis and that the leftover rectangles are thrown away.
A PDF uniform on $x$ when $y=y_0$ and 0 otherwise would give Alice nothing.
In light of @RossMillikan comment
$$P(x,y)=\begin{cases}
\frac{1}{5}&[0,0],[1,0],[0,1],[1,1],[\frac{1}{4},\frac{1}{4}]\\
0&\text{otherwise}
\end{cases}$$
