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Approximate a Lipschitz function by an affine function

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1

$begingroup$

Suppose $f: A rightarrow mathbbR$ is a Lipschitz function (Lipschitz constant $L$), and $A$ is a bounded subset of $mathbbR^n$ (i.e., for any $x in A, ||x||<M$).

If I am going to use an affine function to approximate this Lipschitz function, what is the $ell 2$ norm of their difference? That is, what is

$$sup_f inf_g ||f-g|| = sup_f inf_g int_A (f(x) -g(x))^2 dx$$

$f$ is defined above, and $g$ is an affine function defined on $A$.

real-analysis functional-analysis measure-theory lebesgue-measure

share|cite|improve this question

edited Mar 10 at 21:07

coolcat

asked Mar 10 at 20:55

coolcatcoolcat

1159

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you sure such a supremum is finite ?
  $endgroup$
  – TheSilverDoe
  Mar 10 at 20:59
  

  
  
  
  

add a comment | 

1

$begingroup$

Suppose $f: A rightarrow mathbbR$ is a Lipschitz function (Lipschitz constant $L$), and $A$ is a bounded subset of $mathbbR^n$ (i.e., for any $x in A, ||x||<M$).

If I am going to use an affine function to approximate this Lipschitz function, what is the $ell 2$ norm of their difference? That is, what is

$$sup_f inf_g ||f-g|| = sup_f inf_g int_A (f(x) -g(x))^2 dx$$

$f$ is defined above, and $g$ is an affine function defined on $A$.

real-analysis functional-analysis measure-theory lebesgue-measure

share|cite|improve this question

edited Mar 10 at 21:07

coolcat

asked Mar 10 at 20:55

coolcatcoolcat

1159

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you sure such a supremum is finite ?
  $endgroup$
  – TheSilverDoe
  Mar 10 at 20:59
  

  
  
  
  

add a comment | 

$begingroup$

Suppose $f: A rightarrow mathbbR$ is a Lipschitz function (Lipschitz constant $L$), and $A$ is a bounded subset of $mathbbR^n$ (i.e., for any $x in A, ||x||<M$).

If I am going to use an affine function to approximate this Lipschitz function, what is the $ell 2$ norm of their difference? That is, what is

$$sup_f inf_g ||f-g|| = sup_f inf_g int_A (f(x) -g(x))^2 dx$$

$f$ is defined above, and $g$ is an affine function defined on $A$.

real-analysis functional-analysis measure-theory lebesgue-measure

share|cite|improve this question

edited Mar 10 at 21:07

coolcat

asked Mar 10 at 20:55

coolcatcoolcat

1159

$endgroup$

share|cite|improve this question

edited Mar 10 at 21:07

coolcat

asked Mar 10 at 20:55

coolcatcoolcat

1159

share|cite|improve this question

edited Mar 10 at 21:07

coolcat

asked Mar 10 at 20:55

coolcatcoolcat

1159

asked Mar 10 at 20:55

coolcatcoolcat

1159

asked Mar 10 at 20:55

coolcatcoolcat

1159