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.
Please note that an alternate way of looking at $g$ is the following. Let $Y$ be a random variable having a pdf $(x+(1−x)p)h_1(⋅)+(1−x)(1−p)h_0(⋅)$, and define the random variable $Z$ as $$\frac{(x+(1−x)p)h_1(Y)}{(x+(1−x)p)h_1(Y)+(1−x)(1−p)h_0(Y)}$$ Note that $g(x):=λ+E[f(Z)].
Best Answer
It is not true in general that two concave functions intersect at at most two points. For example, the functions $$ f(x) \;=\; -x^2 \qquad\text{and}\qquad g(x) \;=\; -x^2 + \sin x. $$ are both concave, with $g''(x) = -2 -\sin x < -1$, but the graphs of $f$ and $g$ have infinitely many intersections.