Simple example of a function that is convex but not jointly convex?












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Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?



The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.



Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$



Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.










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  • 4




    $begingroup$
    What about $f(x,y)=xy$?
    $endgroup$
    – A.Γ.
    Jan 5 at 18:47






  • 1




    $begingroup$
    What is a jointly convex function?
    $endgroup$
    – Mostafa Ayaz
    Jan 5 at 18:50
















0












$begingroup$


Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?



The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.



Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$



Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.










share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    What about $f(x,y)=xy$?
    $endgroup$
    – A.Γ.
    Jan 5 at 18:47






  • 1




    $begingroup$
    What is a jointly convex function?
    $endgroup$
    – Mostafa Ayaz
    Jan 5 at 18:50














0












0








0





$begingroup$


Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?



The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.



Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$



Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.










share|cite|improve this question











$endgroup$




Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?



The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.



Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$



Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.







functions convex-analysis






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 5 at 19:28







user1936752

















asked Jan 5 at 18:42









user1936752user1936752

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5321513








  • 4




    $begingroup$
    What about $f(x,y)=xy$?
    $endgroup$
    – A.Γ.
    Jan 5 at 18:47






  • 1




    $begingroup$
    What is a jointly convex function?
    $endgroup$
    – Mostafa Ayaz
    Jan 5 at 18:50














  • 4




    $begingroup$
    What about $f(x,y)=xy$?
    $endgroup$
    – A.Γ.
    Jan 5 at 18:47






  • 1




    $begingroup$
    What is a jointly convex function?
    $endgroup$
    – Mostafa Ayaz
    Jan 5 at 18:50








4




4




$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47




$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47




1




1




$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50




$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50










0






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