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2019 The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$
Taras Banakh, Eliza Jabłońska, Wojciech Jabłoński
Topol. Methods Nonlinear Anal. 54(1): 247-256 (2019). DOI: 10.12775/TMNA.2019.040

Abstract

We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$. Moreover, if $K$ is locally connected and each non-empty open subset of $K$ is not flat, then for any (analytic) non-meager subset $A\subset K$ the sum $A^{+n}$ of $n$ copies of $A$ is not meager in $\mathbb R^n$ (and then the sum $A^{+2n}$ of $2n$ copies of the analytic set $A$ has non-empty interior in $\mathbb R^n$ and the set $(A-A)^{+n}$ is a neighbourhood of zero in $\mathbb R^n$). This implies that a mid-convex function $f\colon D\to\mathbb R$ defined on an open convex subset $D\subset\mathbb R^n$ 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 $D$.

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Taras Banakh. Eliza Jabłońska. Wojciech Jabłoński. "The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$." Topol. Methods Nonlinear Anal. 54 (1) 247 - 256, 2019. https://doi.org/10.12775/TMNA.2019.040

Information

Published: 2019
First available in Project Euclid: 16 July 2019

zbMATH: 07131283
MathSciNet: MR4018279
Digital Object Identifier: 10.12775/TMNA.2019.040

Rights: Copyright © 2019 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.54 • No. 1 • 2019
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