Are minimal pairs of convex sets unique?

[Minimal pair of lenses]


We already know

  1. Let (A,B) be a minimal pair of compact convex sets in X. If X=R or R2 then (A+x,B+x), x[is an element of a set] X are all minimal pairs equivalent to (A,B).

    In R3 there exist two equivalent minimal pairs (A,B) and (C,D) that are not translates of each other:
    A = { (x, y, z) [is an element of] [0,1] 3 | x + y + z [less than or equal to] 2},
    B = { (x, y, z) [is an element of] [0,1] 3 | x + y + z   [greater than or equal to] 1},
    C = { (x, y, z) [is an element of] [0,2] 3 | 2 [less than or equal to] x + y + z [less than or equal to] 3 , x + y [greater than or equal to] 1 , x + z [greater than or equal to] 1 , y + z [greater than or equal to] 1},
    D = { (x, y, z) [is an element of] [0,2] 3 | 3 [less than or equal to] x + y + z [less than or equal to] 4 , x + y [less than or equal to] 3 , x + z [less than or equal to] 3 , y + z [less than or equal to] 3},


  2. Let X=R3,   T1, T2 :R3 [maps] R3,   T1(x,y,z)=(-x,-y,-z), T2(x,y,z)=(-x,-y,z). There exist compact convex sets A and B such that (A,T1A) and (B,T1B) are equivalent minimal pairs that are not translates of each other. Also (A,T2A) and (B,T2B) are equivalent minimal pairs that are not translates of each other.

Do you have a question?
Write to rich@amu.edu.pl ,
lh09@rz.uni-karlsruhe.de




We still do not know

  1. What are all minimal pairs equivalent to (A,B):
    A = { (x, y, z) [is an element of] [0,1] 3 | x + y + z [less than or equal to] 2},
    B = { (x, y, z) [is an element of] [0,1] 3 | x + y + z   [greater than or equal to] 1},


  2. Let   T3 :R3 [maps] R3,   T3(x,y,z)=(x,y,-z). Do there exist compact convex sets A and B such that (A, T3A) and (B, T3B) are equivalent minimal pairs that are not translates of each other?

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 *