Is there a finite set $mathcal{D}$ such that $E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2$...












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Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$



Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.



I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.






real-analysis multivariable-calculus convex-analysis convex-optimization







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    Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
    $$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$



    Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.



    I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.






    real-analysis multivariable-calculus convex-analysis convex-optimization







    share|cite|improve this question



























      1












      1








      1







      Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
      $$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$



      Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.



      I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.






      real-analysis multivariable-calculus convex-analysis convex-optimization







      share|cite|improve this question















      Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
      $$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$



      Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.



      I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.






      real-analysis multivariable-calculus convex-analysis convex-optimization




      real-analysis multivariable-calculus convex-analysis convex-optimization






      share|cite|improve this question















      share|cite|improve this question













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      edited Nov 26 at 9:54

























      asked Nov 26 at 2:50









      Joe Man Analysis

      33419




      33419






















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          Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.






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            Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.






            share|cite|improve this answer


























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              Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.






              share|cite|improve this answer
























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                Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.






                share|cite|improve this answer












                Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 26 at 12:19









                gerw

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                19k11133






























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