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Convexity of a Set

Is there a straightforward way to determine if this set is convex?How can I prove the hypersurface $M$ is neither convex nor concave?How to determine whether a function is concave, convex, quasi-concave and quasi-convexConvexity check sub- / superlevel setsChecking whether a multivariable function is convexShow convexity of a set using the definition of convex setQuasiconcavity of the product functionProving Convexity of Multivariate Function using Conditional Univariate Convexityconvexity/concavity of product of powerslevel set strongly convex and smooth functions

2

$begingroup$

Consider the following function,

$$
f(x, y) = e^m e^-y+n e^-x-x-y left(a x e^y+b e^x y+c x yright)
$$

where $a, b, c, m$ and $n$ are positive constants.

I want to show $f(x, y)$ is quasi-concave. To that end, we define the set $S_alpha subset mathbb R^2$ by,

$$
S_alpha = (x, y)
$$

How can I show the set $S_alpha$ is convex? Is there any simpler way than using the definition $$f(x_1, y_1) > alpha wedge f(x_2, y_2) > alpha Rightarrow f(t x_1+(1-t)x_2, t y_1+(1-t)y_2) > alpha ;text for ; 0 < t < 1?$$ Is it possible to show convexity by composing some basic functions like $xy$, $x e^y$, ...?

calculus multivariable-calculus convex-optimization convex-analysis

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edited Jun 1 '11 at 23:14

Namaste

1

asked Jun 1 '11 at 21:31

HeliumHelium

317215

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am not sure on how to go for the quasi-concave part, but my first guess would be to use derivatives to show your set's convex, would make your life a little easier.
  $endgroup$
  – Patrick Da Silva
  Jun 2 '11 at 5:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I have no idea how can one show the convexity of a set by derivation. Any hint?
  $endgroup$
  – Helium
  Jun 2 '11 at 6:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mohsen The way of showing that $f(x,y)$ is convex by using partial derivatives means you will have to compose a Hessian matrix of $f(x,y)$. But it does not look like it is the most convenient method for this particular function to compute its partial derivatives.
  $endgroup$
  – Koba
  Apr 28 '12 at 5:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not sure this is right. $e^e^-x$, $e^e^-y$, $e^-x$, $e^x$, ... These are all convex functions. Why do you think it'll be quasi-concave?
  $endgroup$
  – Thomas Ahle
  Mar 11 at 20:24
  
  

  
  
  
  

add a comment | 

2

$begingroup$

Consider the following function,

$$
f(x, y) = e^m e^-y+n e^-x-x-y left(a x e^y+b e^x y+c x yright)
$$

where $a, b, c, m$ and $n$ are positive constants.

I want to show $f(x, y)$ is quasi-concave. To that end, we define the set $S_alpha subset mathbb R^2$ by,

$$
S_alpha = (x, y)
$$

How can I show the set $S_alpha$ is convex? Is there any simpler way than using the definition $$f(x_1, y_1) > alpha wedge f(x_2, y_2) > alpha Rightarrow f(t x_1+(1-t)x_2, t y_1+(1-t)y_2) > alpha ;text for ; 0 < t < 1?$$ Is it possible to show convexity by composing some basic functions like $xy$, $x e^y$, ...?

calculus multivariable-calculus convex-optimization convex-analysis

share|cite|improve this question

edited Jun 1 '11 at 23:14

Namaste

1

asked Jun 1 '11 at 21:31

HeliumHelium

317215

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am not sure on how to go for the quasi-concave part, but my first guess would be to use derivatives to show your set's convex, would make your life a little easier.
  $endgroup$
  – Patrick Da Silva
  Jun 2 '11 at 5:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I have no idea how can one show the convexity of a set by derivation. Any hint?
  $endgroup$
  – Helium
  Jun 2 '11 at 6:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mohsen The way of showing that $f(x,y)$ is convex by using partial derivatives means you will have to compose a Hessian matrix of $f(x,y)$. But it does not look like it is the most convenient method for this particular function to compute its partial derivatives.
  $endgroup$
  – Koba
  Apr 28 '12 at 5:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not sure this is right. $e^e^-x$, $e^e^-y$, $e^-x$, $e^x$, ... These are all convex functions. Why do you think it'll be quasi-concave?
  $endgroup$
  – Thomas Ahle
  Mar 11 at 20:24
  
  

  
  
  
  

add a comment | 

$begingroup$

Consider the following function,

$$
f(x, y) = e^m e^-y+n e^-x-x-y left(a x e^y+b e^x y+c x yright)
$$

where $a, b, c, m$ and $n$ are positive constants.

I want to show $f(x, y)$ is quasi-concave. To that end, we define the set $S_alpha subset mathbb R^2$ by,

$$
S_alpha = (x, y)
$$

How can I show the set $S_alpha$ is convex? Is there any simpler way than using the definition $$f(x_1, y_1) > alpha wedge f(x_2, y_2) > alpha Rightarrow f(t x_1+(1-t)x_2, t y_1+(1-t)y_2) > alpha ;text for ; 0 < t < 1?$$ Is it possible to show convexity by composing some basic functions like $xy$, $x e^y$, ...?

calculus multivariable-calculus convex-optimization convex-analysis

share|cite|improve this question

edited Jun 1 '11 at 23:14

Namaste

1

asked Jun 1 '11 at 21:31

HeliumHelium

317215

$endgroup$

share|cite|improve this question

edited Jun 1 '11 at 23:14

Namaste

1

asked Jun 1 '11 at 21:31

HeliumHelium

317215

share|cite|improve this question

edited Jun 1 '11 at 23:14

Namaste

1

asked Jun 1 '11 at 21:31

HeliumHelium

317215

edited Jun 1 '11 at 23:14

Namaste

1

edited Jun 1 '11 at 23:14

Namaste

1

asked Jun 1 '11 at 21:31

HeliumHelium

317215

asked Jun 1 '11 at 21:31

HeliumHelium

317215