daniel j. greenhoe

By the definition of convexity, the most general space in which we can
define a convex set is a vector space; That is, a set D is convex in X
if it satisfies
x,y in D ==> ax + (1-a)y in D forall a in (0,1)
And because this definition requires scalar-vector multiplication and
vector-vector addition, we at least need a vector space to define
convexity.

But my question is, is there any generalization to convexity that does
_not_ require a vector space? Any generalized definition that only
requires a metric space or even just a topological space?

I know that there are generalized definitions for orthogonality that
do not require an inner-product space, but only require a normed
vector space. This might make someone wonder if there is any kind of
work done to generalize convexity also.

Anyone have any example of such a thing?

Many thanks in advance,
Dan Greenhoe

alex.lupas

Daniel J. Greenhoe a scris:


Hmm...Interesting !

I remember following fact: let a be fixed, namely a=1/2 .
A set D with the property : x,y in D ===> (x+y)/2 in D
is called a "perfect set".
Regarding "perfect sets: there is a book published in Suisse
(around 1950-1860) by Sophie Picard "Sur les ensembles parfait(s)",
Geneve {?} [perhaps her thesis , supervisor Jovan Karamata (?)].
Perhaps you find some informations in the book by
Lars Hormander"Notions of Convexity",Birkhauser,1994
ISBN0-8176-32-99-0

stephen j. herschkorn

I posed a similar question here once. See the thread, "General
convexity and continuity," which I started on 31 December 2002.

--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan

andrej dujella

In axiomatic theory of convexity
(http://en.wikipedia.org/wiki/Convex_set#Abstract_.28axiomatic.29_convexity
http://www.math.bgu.ac.il/~kubis/thesis.html),
convexity over a set X is a collection C of subsets of X, satisfying the
axioms:
1. the empty set and X are in C;
2. C is closed under intesections;
3. The union of a chain of elements in C is in C.
Elements of C are called convex sets.

Good references:

M. van de Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993.
V.P. Soltan, Introduction to the Axiomatic Theory of Convexity, (Russian)
Shtinca, Kishinev, 1984.
R.E. Jamison, A General Theory of Convexity, Dissertation, University of
Washington, Seattle, 1974.
M. van de Vel, Pseudo-boundaries and pseudo-interiors for topological
convexities, Dissert. Math. 210 (1983), 1-72.
M. van de Vel, Finite dimensional convex structures I: general results, Top.
Appl. 14 (1982), 201-225.

daniel j. greenhoe

My understanding of "perfect sets" (in topology) is a set that is
equal to its derived set; meaning that the set is closed and has no
isolated points. In my thinking anyway, this would not the same
concept as convexity. But I would be happy to be proven wrong ^__^


I see this book is in my school's library --- I will borrow it from
there. It looks like a very interesting reference. Thank you very
much.

Dan Greenhoe

[ Moderator's note: Alex's usage of "perfect set" is
completely different from the usual topological definition,
and I've never seen it before (which, of course, doesn't
mean it doesn't exist). A look at the index of the
Hormander book and a search using Amazon.com's "Search inside
the book" didn't find any mention of "perfect".
-RI ]

daniel j. greenhoe

I should have checked Wikipedia before posting. Thank you very much
for so many great references. They should keep me busy for some time.

Dan Greenhoe

bo198214

There is also a notion of convexity in order theory.
For a partially ordered set X a subset A is called convex if for any
a<=b in A also all x with a<=x<=b are in A (in other words the closed
interval [a,b] lies in A).

daniel j. greenhoe

I have found this thread at
http://groups.google.com/group/sci.math/browse_thread/thread/346d0b11d01f078/582bfda991278cfa
There appears to be a wealth of information there including
references. Thank you so much!

Dan Greenhoe

daniel j. greenhoe

Interesting! And very cool. Thanks so much.

Using Google Book Search, I did find a book reference that appears to
maybe have this definition; but I am not sure about the author's
definition of [a, b]:
General topology By Mangesh Ganesh Murdeshwar, page 18
http://books.google.com/books?vid=ISBN8122402461
http://www.worldcat.org/oclc/46120146
http://www.amazon.com/dp/8122402461

I would greatly appreciate any other references that you think are
good.

Dan Greenhoe

gowan4

In axiomatic geometry there is a notion of betweenness for points (on
a line). In such a geometry a set S could be convex if whenever x and
y are in S and z is between x and y then z is in S. No doubt this is
covered in some of the other references. I think there is a notion of
convexity for sets in a lattice having certain properties. Sorry not
to be able to give more details.