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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)].

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I also have a question related. What if two concave , increasing, continuous functions; can we claim they intersect at most finitely may times in an interval? Regards, D. –  daria Sep 1 '12 at 17:56
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@daria: First, this is not an answer. Secon, do not kidnap others' questions to aks yours. Instead, ask a new question in a new thread and show your efforts, ideas, background there. You should erase this one. –  DonAntonio Sep 2 '12 at 10:16
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@DonAntonio: I have moved this non-answer to a comment so that daria can pose it as a new question and can read your comment. –  robjohn Sep 2 '12 at 20:35

1 Answer 1

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.

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Hi Jim Thanks. So, that comes to proving or disproving the number of intersections for the related concave functions. 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 $g(x):=λ+E[f(Z)]$ –  Premkumar K. May 31 '11 at 17:23

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