positive linear combination of quasi-concave functions












0














I have a question that I cannot manage to get around. I need to answer the following:




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.




Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?



If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.



Thanks in advance, any help would be useful






convex-analysis







share|cite|improve this question



























    0














    I have a question that I cannot manage to get around. I need to answer the following:




    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.




    Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?



    If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.



    Thanks in advance, any help would be useful






    convex-analysis







    share|cite|improve this question

























      0












      0








      0







      I have a question that I cannot manage to get around. I need to answer the following:




      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.




      Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?



      If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.



      Thanks in advance, any help would be useful






      convex-analysis







      share|cite|improve this question













      I have a question that I cannot manage to get around. I need to answer the following:




      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.




      Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?



      If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.



      Thanks in advance, any help would be useful






      convex-analysis




      convex-analysis






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 9 at 15:29









      user20105

      82




      82






















          1 Answer
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          0














          The function $xmapsto sqrt |x|$ 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 $f$.
          On the interval $(-1,1)$ the function $f$ reduces to
          $$
          f(x) = a sqrt{1-x} + b sqrt{x+1},
          $$
          which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.



          Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
          Then the sub-level set
          $$
          left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
          $$
          contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.



          Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.






          share|cite|improve this answer























          • Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
            – user20105
            Dec 13 at 14:56










          • I expanded the answer. Could you be more specific? What argument is difficult to understand?
            – daw
            Dec 13 at 15:06










          • I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
            – user20105
            Dec 13 at 15:31










          • The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
            – daw
            Dec 13 at 15:41










          • I see it! Thank you for the answer and the explanation!
            – user20105
            Dec 13 at 15:47













          Your Answer





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          1 Answer
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          1 Answer
          1






          active

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          active

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          active

          oldest

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          0














          The function $xmapsto sqrt |x|$ 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 $f$.
          On the interval $(-1,1)$ the function $f$ reduces to
          $$
          f(x) = a sqrt{1-x} + b sqrt{x+1},
          $$
          which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.



          Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
          Then the sub-level set
          $$
          left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
          $$
          contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.



          Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.






          share|cite|improve this answer























          • Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
            – user20105
            Dec 13 at 14:56










          • I expanded the answer. Could you be more specific? What argument is difficult to understand?
            – daw
            Dec 13 at 15:06










          • I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
            – user20105
            Dec 13 at 15:31










          • The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
            – daw
            Dec 13 at 15:41










          • I see it! Thank you for the answer and the explanation!
            – user20105
            Dec 13 at 15:47


















          0














          The function $xmapsto sqrt |x|$ 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 $f$.
          On the interval $(-1,1)$ the function $f$ reduces to
          $$
          f(x) = a sqrt{1-x} + b sqrt{x+1},
          $$
          which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.



          Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
          Then the sub-level set
          $$
          left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
          $$
          contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.



          Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.






          share|cite|improve this answer























          • Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
            – user20105
            Dec 13 at 14:56










          • I expanded the answer. Could you be more specific? What argument is difficult to understand?
            – daw
            Dec 13 at 15:06










          • I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
            – user20105
            Dec 13 at 15:31










          • The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
            – daw
            Dec 13 at 15:41










          • I see it! Thank you for the answer and the explanation!
            – user20105
            Dec 13 at 15:47
















          0












          0








          0






          The function $xmapsto sqrt |x|$ 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 $f$.
          On the interval $(-1,1)$ the function $f$ reduces to
          $$
          f(x) = a sqrt{1-x} + b sqrt{x+1},
          $$
          which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.



          Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
          Then the sub-level set
          $$
          left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
          $$
          contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.



          Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.






          share|cite|improve this answer














          The function $xmapsto sqrt |x|$ 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 $f$.
          On the interval $(-1,1)$ the function $f$ reduces to
          $$
          f(x) = a sqrt{1-x} + b sqrt{x+1},
          $$
          which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.



          Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
          Then the sub-level set
          $$
          left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
          $$
          contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.



          Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 13 at 15:06

























          answered Dec 11 at 8:03









          daw

          24k1544




          24k1544












          • Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
            – user20105
            Dec 13 at 14:56










          • I expanded the answer. Could you be more specific? What argument is difficult to understand?
            – daw
            Dec 13 at 15:06










          • I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
            – user20105
            Dec 13 at 15:31










          • The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
            – daw
            Dec 13 at 15:41










          • I see it! Thank you for the answer and the explanation!
            – user20105
            Dec 13 at 15:47




















          • Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
            – user20105
            Dec 13 at 14:56










          • I expanded the answer. Could you be more specific? What argument is difficult to understand?
            – daw
            Dec 13 at 15:06










          • I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
            – user20105
            Dec 13 at 15:31










          • The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
            – daw
            Dec 13 at 15:41










          • I see it! Thank you for the answer and the explanation!
            – user20105
            Dec 13 at 15:47


















          Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
          – user20105
          Dec 13 at 14:56




          Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
          – user20105
          Dec 13 at 14:56












          I expanded the answer. Could you be more specific? What argument is difficult to understand?
          – daw
          Dec 13 at 15:06




          I expanded the answer. Could you be more specific? What argument is difficult to understand?
          – daw
          Dec 13 at 15:06












          I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
          – user20105
          Dec 13 at 15:31




          I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
          – user20105
          Dec 13 at 15:31












          The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
          – daw
          Dec 13 at 15:41




          The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
          – daw
          Dec 13 at 15:41












          I see it! Thank you for the answer and the explanation!
          – user20105
          Dec 13 at 15:47






          I see it! Thank you for the answer and the explanation!
          – user20105
          Dec 13 at 15:47




















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