A proper, lower semicontinuous, convex function with no subgradient?












1












$begingroup$


Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.




Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?




If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?






real-analysis functional-analysis convex-analysis convex-optimization







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.




    Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?




    If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?






    real-analysis functional-analysis convex-analysis convex-optimization







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.




      Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?




      If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?






      real-analysis functional-analysis convex-analysis convex-optimization







      share|cite|improve this question









      $endgroup$




      Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.




      Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?




      If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?






      real-analysis functional-analysis convex-analysis convex-optimization




      real-analysis functional-analysis convex-analysis convex-optimization






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 31 at 16:44









      BigbearZzzBigbearZzz

      9,01521652




      9,01521652






















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          $begingroup$

          The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
          (See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)






          share|cite|improve this answer









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            $begingroup$

            The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
            (See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
              (See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
                (See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)






                share|cite|improve this answer









                $endgroup$



                The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
                (See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 31 at 18:28









                RigelRigel

                11.4k11320




                11.4k11320






























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