How did we get interested in pairs of convex sets?

[Jerzy Grzybowski] [Diethard Pallaschke] [Ryszard Urbański]
Jerzy Grzybowski Diethard Pallaschke Ryszard Urbański
jgrz@amu.edu.pl lh09@rz.uni-karlsruhe.de rich@amu.edu.pl

The investigation of minimal pairs of compact convex sets has been stimulated by the quasidifferential calculus of V. F. Demyanov and A. M. Rubinov [7] which is based on formal rules for pairs of subdifferentials.

Due to the formal rules of this calculus, it may happen that already after a few operations the pairs of subdifferential increase so fast that they become unwiedly for the practical use of the calculus. Here the natural question on the existence and charactrization of minimal pairs of compact convex sets arises.

The first results in this direction have been presented by Ryszard Urbański and Diethard Pallaschke in summer 1989, in the seminar of W. Orlicz in Poznań (see [20], [24]). In particularly algebraic and geometric methods for the characterization of minimality were developed in [24] where several examples are presented, as for instance pairs of lenses and star of David (pair of crossed triangles).

Next it was shown independently by Jerzy Grzybowski [10] and Stefan Scholtes [33] that minimal pairs in the plane are unique up to translations; furtheron Jerzy Grzybowski [10] gave the first counterexample that this result is not true in higher dimensions.

Based on this results started the systematic investigation of minimal pairs of bounded or compact convex sets.






Minimal pair of lenses



[minimal pair of lenses]

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"Star of David"



[star of David]

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* 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 *
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