How do minimal pairs of convex sets
relate to fractions?


[The family of closed convex polygons]
The family S of closed convex polygons (including the degenerate ones) in R 2 with one-dimensiond faces parallel to intervals D,E,F and G.


We already know

Let ( S, . ) be a commutative semigroup ordered by the relation [less than or equal to]. Moreover, let S satisfies the order law of cancellation.

if asis less than or equal tobs for some s [is an element of a set]S, then a[is less than or equal to]b

If a and b[is an element of a set]S, then by fraction a / b we mean the pair (a,b) [is an element of a set]S 2.
The set S 2 may be ordered by the relation: a'/b' [partial order] a/b if and only if a'[is less than or equal to]a and b'[is less than or equal to]b.
The set S may have following properties:
  1. For all a,b [is an element of a set]S there exist   min(a,b) and   max(a,b)
  2. For all a, b, c [is an element of a set]S    a + max(b,c) = max(a + b, a + c)
  3. For all a,b[is an element of a set]S there exist minimal fraction c / d
Properties 1, 2 and 3 are satisfied by ( S, ., [is less than or equal to] ) = ( N, ., | ).
Properties 1, 2 and 3 are satisfied by ( K(X), +, [subset] ).
Properties 1 and 3 are satisfied by ( B(X), [Minkowski sum], [subset] ).
Do you have a question?
Write to rich@amu.edu.pl ,
lh09@rz.uni-karlsruhe.de




We still do not know

What are essential common properties of ( N, ., | ) , ( K(X), +, ) and ( B(X), [Minkowski sum], [subset] ) ?
What are essential differences between ( N, ., | ) , ( K(X), +, [subset] ) and ( B(X), [Minkowski sum], [subset] ) ?
What are other examples of ( S, ., [is less than or equal to] ) that have essential properties of ( N, ., | ) , ( K(X), +, [subset] )?
Do you have an answer?
Write to rich@amu.edu.pl,
lh09@rz.uni-karlsruhe.de


* Do minimal pairs of convex sets exist? * Are minimal pairs of compact sets unique? *
* What about convex polytops? * What are invariants of minimal pairs of convex sets? *
* How do minimal pairs of convex sets relate to fractions? *
* History of minimal pairs of convex sets *
* How did we get interested in pairs of convex sets? *
* References * Definitions *
* Home *