Boundedness and continuity of additive and convex functionals

We prove that for a continuum K ⊂ R n the sum K +n of n copies of K has non-empty interior in R n if and only if K is not flat in the sense that the affine hull of K coincides with R n . Moreover, if K is locally connected and each non-empty open subset of K is not flat, then for any (analytic) non-meager subset A ⊂ K the sum A +n of n copies of A is not meager in R n (and then the sum A +2n of 2n copies of the analytic set A has non-empty interior in R n and the set (A − A) +n is a neighborhood of zero in R n ). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ R n is continuous if it is upper bounded on some non-flat continuum in D or on a non-meager analytic subset of a locally connected nowhere flat subset of D.1991 Mathematics Subject Classification. 26B05, 26B25, 54C05, 54C30, 54D05.

show abstract

“…On the other hand, Ger and Kominek [7] proved that A(X) = B(X) for any Baire topological vector space X.…”

Section: It Is Clear Thatmentioning

confidence: 99%

The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$

Банах¹,

Jabłońska²,

Jabłoński³

2019

TMNA

View full text Add to dashboard Cite

show abstract

“…He is also very active in submitting papers concerning the applications of functional equations or inequalities to functional analysis or harmonic analysis (let us mention e.g. [3,8,11,13,15], or [16], just to cite a few).…”

Section: Maciej Sablikmentioning

confidence: 99%

Roman Ger, 70 years of a mathematician’s adventurous life

Sablik

2016

Aequat. Math.

View full text Add to dashboard Cite

“…One can put the above definition into a more general setting (e.g., for topological groups); however, with respect to our main result, we restrict ourselves to considering this class for X and Y being inner product spaces. Such a class B (for particular X and Y ) was considered by Kuczma [11], [12] p. 206 ff, Ger-Kuczma [8], Ger-Kominek [7]. A characterization of B(R n , R) was given by Smital [16].…”

Section: Boundedness and Continuitymentioning

confidence: 99%

Orthogonality equation almost everywhere

Chmieliński¹,

Rätz²

1998

Publ. Math. Debrecen

View full text Add to dashboard Cite

For inner product spaces X and Y , we consider the orthogonality equationas well as its restricted versionswhere f : X → Y and the sets M and U are, in some sense, small in X 2 and X, respectively. Under some additional assumptions we prove that for a solution f of the orthogonality equation postulated almost everywhere (in one of the two senses mentioned above) there exists a unique solution f * of the (unrestricted) orthogonality equation such that f and f * are equal almost everywhere on X.

show abstract

Boundedness and continuity of additive and convex functionals

Cited by 14 publications

References 8 publications

The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$

The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$

Roman Ger, 70 years of a mathematician’s adventurous life

Orthogonality equation almost everywhere

Contact Info

Product

Resources

About