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Results of Tadeusz Świątkowski on algebraic sums of sets and their applications in the theory of subadditive functions

Matkowski J.

Journal of Applied Analysis

2012

Vol. 18, nr 1

97-116

A refinement of Steinhaus' theorem on the algebraic sum of subsets of R due to Raikov (1939) was not known to the mathematical community and still is not popular. In 1994, Tadeusz Świątkowski, being not aware of the existence of Raikov's theorem, proved another result of this type. Unfortunately, a few days later he passed away. In this paper we present the theorems of Świątkowski and Raikov and we apply them in the theory of subadditive type inequalities. An improvement of a converse of Minkowski's inequality theorem is presented.

Subadditive periodic functions

Matkowski J.

Opuscula Mathematica

2011

Vol. 31, no. 1

75-96

Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function ƒ0 : [0, 1) ? R is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.