Give an example to $2$ quasi-concave functions on an interval such that any positive linear combination of these two functions is not quasi-concave.

Aryanna Blake
2022-10-23
Answered

Give an example to $2$ quasi-concave functions on an interval such that any positive linear combination of these two functions is not quasi-concave.

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scranna0o

Answered 2022-10-24
Author has **16** answers

The function $x\mapsto \sqrt{|}x|$ is quasi-convex. Let me show that the function

$$f(x)=a\sqrt{|x-1|}+b\sqrt{|x+1|}$$

is not quasi-convex for all $a,b>0$.

The points $x=-1$ and $x=1$ are local minima of $(-1,1)$. On the interval $(-1,1)$the function $(-1,1)$ reduces to

$$f(x)=a\sqrt{1-x}+b\sqrt{x+1},$$

which is a strictly concave function. Hence, $f$ has a local maximum ${x}^{\ast}\in (-1,1)$ with $f({x}^{\ast})>max(f(-1),f(1))$.

which is a strictly concave function. Hence, $f$ has a local maximum ${x}^{\ast}\in (-1,1)$ with $f({x}^{\ast})>max(f(-1),f(1))$.

Now let me choose a sub-level set that contains $1$ and $-1$ but not ${x}^{\ast}$. Then the sub-level set

$$\{x:f(x)\le \frac{f({x}^{\ast})+max(f(-1),f(1))}{2}\}$$

contains $-1,1$ but not ${x}^{\ast}$. Hence this level set is not convex, and $f$ is not quasi-convex.

Note that $-f$$-f$ is not quasi-concave, but is the sum of two quasi-concave functions.

$$f(x)=a\sqrt{|x-1|}+b\sqrt{|x+1|}$$

is not quasi-convex for all $a,b>0$.

The points $x=-1$ and $x=1$ are local minima of $(-1,1)$. On the interval $(-1,1)$the function $(-1,1)$ reduces to

$$f(x)=a\sqrt{1-x}+b\sqrt{x+1},$$

which is a strictly concave function. Hence, $f$ has a local maximum ${x}^{\ast}\in (-1,1)$ with $f({x}^{\ast})>max(f(-1),f(1))$.

which is a strictly concave function. Hence, $f$ has a local maximum ${x}^{\ast}\in (-1,1)$ with $f({x}^{\ast})>max(f(-1),f(1))$.

Now let me choose a sub-level set that contains $1$ and $-1$ but not ${x}^{\ast}$. Then the sub-level set

$$\{x:f(x)\le \frac{f({x}^{\ast})+max(f(-1),f(1))}{2}\}$$

contains $-1,1$ but not ${x}^{\ast}$. Hence this level set is not convex, and $f$ is not quasi-convex.

Note that $-f$$-f$ is not quasi-concave, but is the sum of two quasi-concave functions.

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