# Intersection of two (related) concave functions

Question: In general, two concave functions intersect at at most two points. True or False? If false, can you please provide an example. If true, can you please provide a proof.

Proving or disproving may be hard for any two concave functions. But, the functions I have are related. I am explaining my problem below: (please note that though the functions are related and are known to be concave, the explicit form of the functions are unknown).

Consider the concave functions $f:[0,1] \to \mathbb{R}_+$ and $g:[0,1] \to \mathbb{R}_+$, which are related as follows.

Let $p \in (0,1)$, and let $h_0$ and $h_1$ are Gaussian probability density functions with mean $\mu_0$ and $\mu_1$ respectively, and having the same variance (say $\sigma^2$). $$g(x)= \lambda + \int_{-\infty}^{\infty} f\left(\frac{(x+(1-x)p)h_1(y)}{(x+(1-x)p)h_1(y)+(1-x)(1-p)h_0(y)}\right) \ \left[(x+(1-x)p)h_1(y)+(1-x)(1-p)h_0(y)\right] \ dy$$ where $\lambda > 0$ is some constant. Please note that $g$ is defined by an appropriate Expectation operation on $f(\cdot)$ (i.e., wrt to the pdf $(x+(1-x)p)h_1(y)+(1-x)(1-p)h_0(y)$).

Question: Can we prove that $\{x:g(x) \leq f(x)\}$ is a convex set?

Assume/Given: $g(0) > f(0)$ and $g(1) > f(1)$. You can assume smoothness, like twice continuously differentiable, strictly concave etc. But, please do not restrict to monotone functions. But, even if there is a solution/counter-example to monotone functions, it would be great.