Let $f:[0,1]^2 \rightarrow \{0,1\}$, $f_B(b) = \int_0^1 f(b, s)\; ds$ and $f_S(s) = \int_0^1 f(b, s)\; db$, such that $f_B$ is non-decreasing and $f_S$ is non-increasing. What can we infer about $f$ from the monotonicity of $f_B$ and $f_S$?
For instance, the following class of functions satisfies the monotonicity constraints \begin{align*} f(b,s) = \begin{cases} 1 & s \leq g(b)\\ 0 & otherwise \end{cases} \end{align*} for some non-decreasing function $g$.
Another class of functions which satisfies the monotonicity constraints can be constructed using the set
$A = \{(b, s) \in [0,1]^2\mid g_1(b) \geq s \geq g_2(b), g_1'(b) = g_2'(b) >0, g_2(b)\leq g_1(b)\leq 1, g_1(0) \leq 0\}$
with \begin{align*} f(b,s) = \begin{cases} 1 & (b,s) \in A\\ 0 & otherwise. \end{cases} \end{align*}
What properties does $f$ have? Is there even a way to fully characterize the class of functions $f$ satisfying the monotonicity constraints on $f_B$ and $f_S$?
There seems to be a connection to (algebraic) topology through the set $B = \{(b, s)\in [0,1]^2\mid f(b, s)=1\}$, too.