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Translative covering of a square by a sequence of arbitrary-oriented squares: Part II

Januszewski J.

Demonstratio Mathematica

2008

Vol. 41, nr 2

441-454

Given a collection of squares of arbitrary side lengths whose total area is at least 3. Then the unit square can be covered by translates of these squares.

Translative packing two squares in the unit square

Januszewski J.

Demonstratio Mathematica

2003

Vol. 36, nr 3

757--762

Any two squares with total area not greater than 0.4 can be translatively packed in the unit square. The translative packing two squares lying in parallel positions is also considered.

Extension theorems in axiomatic theory of convexity

Kubiś W.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

89-96

We present a criterion for extending convexity preserving maps of convexity spaces. In a special case of convexity generated by a lattice structure this gives Sikorski's Extension Criterion for extending of maps of lattices. We also consider the class of convexity absolute extensors. It appears that complete Boolean algebras with a natural convexity belong to this class. In particular, we present an analogue of Tietze-Urysohn's Extension Theorem for maps of convexity spaces with values in a complete Boolean algebra.

On common summands and anti-summands of compact convex sets

Grzybowski J., Urbański A., Wiernowolski M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 1

69-76

In this paper we examine families of common summands and anti-summands of compact convex sets. In particular, we prove the equivalence of minimality problems for common anti-summands and minimal pairs of compact convex sets. We also give partial answers to the question whether minimal anti-summands of polytopes are polytopes as well.