Convex set

OReilly · #1 $Old$ August 10th, 2006, 08:14 AM

Is set $\left\{ {A,B} \right\}$ , $A \ne B$ convex?

Since there is no AB segment in set then set is concave.

Am I right?

CaptainBlack · #2 $Old$ August 10th, 2006, 08:43 AM

Quote:

Originally Posted by OReilly

Is set $\left\{ {A,B} \right\}$ , $A \ne B$ convex?

Since there is no AB segment in set then set is concave.

Am I right?

Is $\left\{ {A,B} \right\}$ a subset of a vector space?

Does it not have to be to be able to ask the question.

If it is; it does not contain any point on the segment $AB$
if $A \ne B$

RonL

OReilly · #3 $Old$ August 10th, 2006, 11:43 AM

Quote:

Originally Posted by CaptainBlack

Is $\left\{ {A,B} \right\}$ a subset of a vector space?

Does it not have to be to be able to ask the question.

If it is; it does not contain any point on the segment $AB$
if $A \ne B$

RonL

I don't know, I retyped question from book.

In the book there isn't definition of convex set, so I have read definition from one internet site which says that convex set is "a set of points in which all segments connecting points of the set lie entirely in the set".

I didn't understand the question quite so I don't know.

CaptainBlack · #4 $Old$ August 10th, 2006, 12:06 PM

Quote:

Originally Posted by OReilly

I don't know, I retyped question from book.

In the book there isn't definition of convex set, so I have read definition from one internet site which says that convex set is "a set of points in which all segments connecting points of the set lie entirely in the set".

I didn't understand the question quite so I don't know.

A convex set is usualy a subset of a vector space such that for any two
points in the subset all the points between are also in the subset.

RonL

OReilly · #5 $Old$ August 10th, 2006, 12:16 PM

Quote:

Originally Posted by CaptainBlack

A convex set is usualy a subset of a vector space such that for any two
points in the subset all the points between are also in the subset.

RonL

So question from book is not well formulated? Incomplete?

JakeD · #6 $Old$ August 10th, 2006, 02:53 PM

Quote:

Originally Posted by CaptainBlack

A convex set is usualy a subset of a vector space such that for any two
points in the subset all the points between are also in the subset.

Quote:

Originally Posted by OReilly

In the book there isn't definition of convex set, so I have read definition from one internet site which says that convex set is "a set of points in which all segments connecting points of the set lie entirely in the set".

Aren't these two definitions saying the same thing in different words?

Quote:

Originally Posted by OReilly

Is set $\left\{ {A,B} \right\}$ , $A \ne B$ convex?

Since there is no AB segment in set then set is concave.

Am I right?

Yes, except the word "concave" does not apply. The set is not convex.

CaptainBlack · #7 $Old$ August 10th, 2006, 03:04 PM

Quote:

Originally Posted by JakeD

Aren't these two definitions saying the same thing in different words?

Yes, except the word "concave" does not apply. The set is not convex.

No, one specifies the existence of a vector space, and the other leaves
us wondering about what it might mean by a segment for an arbitrary
space, and would the definition of convexity work if there were no segment.

Or maybe I am over analysing the question.

Anyway, if they are elements of a vector space there is no problem, the set is
not convex.

RonL

OReilly · #8 $Old$ August 10th, 2006, 05:40 PM

Quote:

Originally Posted by JakeD

Aren't these two definitions saying the same thing in different words?

Yes, except the word "concave" does not apply. The set is not convex.

Isn't terms "concave" and "not convex" same, synonims?

CaptainBlack · #9 $Old$ August 10th, 2006, 11:31 PM

Quote:

Originally Posted by OReilly

Isn't terms "concave" and "not convex" same, synonims?

I think it still requires that the set be a subset of a vector space, but
otherwise it is used in that way.

RonL

JakeD · #10 $Old$ August 10th, 2006, 11:36 PM

Quote:

Originally Posted by OReilly

Isn't terms "concave" and "not convex" same, synonims?

I googled concave set and it appears you are correct and I am not. But then look at this page. It defines concavity the same as convexity (not the opposite of convexity). Since I had never heard of a concave set while I am very familiar with convex sets (which are frequently used in economics), my conclusion is that convexity is a well-established definition while concavity applied to a set is newer and less well-established. Thanks for pointing out my error.

CaptainBlack · #11 $Old$ August 11th, 2006, 12:22 AM

Quote:

Originally Posted by JakeD

I googled concave set and it appears you are correct and I am not. But then look at this page. It defines concavity the same as convexity (not the opposite of convexity). Since I had never heard of a concave set while I am very familiar with convex sets (which are frequently used in economics), my conclusion is that convexity is a well-established definition while concavity applied to a set is newer and less well-established. Thanks for pointing out my error.

However prestigious the institution that the author hails from his usage is
wrong. The usual usage is that a set which is not convex is concave (a
usage that I don't like but there it is) He seems to think concavity and
convexity are the same thing which is confusing (and if they were defined
that way would be a waste of a good word), and then uses the non-standard
one as his default.

RonL

Rebesques · #12 $Old$ August 11th, 2006, 10:37 AM

...The title of that page goes "Concavity (Convexity)"!

Strange... What about defining "Smoker (Passive Smoker)"??
Is this the case where opposites mix, or just a slip of the fingers on the keyboard?

ps ...Well, if concavity and convexity were exactly the same thing, there would be no need to have a definition in the first place.

JakeD · #13 $Old$ August 11th, 2006, 01:45 PM

How and whether you define concavity of a set is important in economics.

The problem with defining concavity of sets as non-convexity is that this can lead to confusion when defining functions as concave or quasi-concave, which are very important in economics. Functions which are concave or quasi-concave have nice maximums.

A function is concave if for all $t \in (0,1),$ $f(tx + (1-t)y) \ge tf(x)+(1-t)f(y).$ Geometrically, that means the set of points below the graph of $f$ is convex. So concavity (non-convexity) of a set has no relation to concavity of a function.

Similarly a function is quasi-concave if the upper level sets $\{x | f(x) \ge y\}$ are convex. Again concavity (non-convexity) of a set has no relation to quasi-concavity of a function.

When teaching these definitions, the important concepts are convexity of set and concavity and quasi-concavity of functions. There is no need to bring in concavity of a set. It just leads to confusion. If you want to say a set is not convex, just say that or use the word non-convex.

What the page I referred to above did was to define concave set in a way that was useful for defining concave functions. At the time it was written (1983), this may have not been incorrect because I don't think the definition of concavity of a set was well-established then. Definitions change over time in mathematics.

Here is a quote from another page google turns up:

I used words like "convex set" and "concave set" in class. It has been pointed out that while "concave set" is a mathematical concept, it is not nearly as commonly used as "convex set", which is the common vocab including in economic applications.

There's good reason for this: In a convex set, if you join 2 points by a line segment, the line segment lies entirely within the set. In a nonconvex (concave) set, if you join 2 points, the line segment may or may not lie within the set, depending on the points you chose. This makes convexity easier to deal with.

He then goes on to discuss quasi-concave functions.