Real value function inequality

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Let $H:mathcalKrightarrow mathcalK$ be a function and $mathcalK$ be a closed compact set. I know that $H$ is non-expansive, e.g.,



$qquad qquad qquad qquad Vert H(x) - H(y) Vert leq Vert x - y Vert.$



Now, I would like to know that under what circumstances the following inequality holds:



$qquad qquad qquad qquad (x-y)^T(H(x) - H(y)) geq Vert H(x) - H(y) Vert^2.$



Thanks in advance!




real-analysis




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    Let $H:mathcalKrightarrow mathcalK$ be a function and $mathcalK$ be a closed compact set. I know that $H$ is non-expansive, e.g.,



    $qquad qquad qquad qquad Vert H(x) - H(y) Vert leq Vert x - y Vert.$



    Now, I would like to know that under what circumstances the following inequality holds:



    $qquad qquad qquad qquad (x-y)^T(H(x) - H(y)) geq Vert H(x) - H(y) Vert^2.$



    Thanks in advance!




    real-analysis




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $H:mathcalKrightarrow mathcalK$ be a function and $mathcalK$ be a closed compact set. I know that $H$ is non-expansive, e.g.,



      $qquad qquad qquad qquad Vert H(x) - H(y) Vert leq Vert x - y Vert.$



      Now, I would like to know that under what circumstances the following inequality holds:



      $qquad qquad qquad qquad (x-y)^T(H(x) - H(y)) geq Vert H(x) - H(y) Vert^2.$



      Thanks in advance!




      real-analysis




      share|cite|improve this question











      Let $H:mathcalKrightarrow mathcalK$ be a function and $mathcalK$ be a closed compact set. I know that $H$ is non-expansive, e.g.,



      $qquad qquad qquad qquad Vert H(x) - H(y) Vert leq Vert x - y Vert.$



      Now, I would like to know that under what circumstances the following inequality holds:



      $qquad qquad qquad qquad (x-y)^T(H(x) - H(y)) geq Vert H(x) - H(y) Vert^2.$



      Thanks in advance!





      real-analysis





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









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      Majid Mohammadi

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