Description

• Title:
  The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb R^n$
• Creator:
  Banakh, Taras;
  Jabłońska, Eliza;
  Jabłoński, Wojciech
• Description:
  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 $D$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2019 Topological Methods in Nonlinear Analysis
• Date:
  2019.07.13
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 54, No 1 (September 2019); 247 - 256;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.040/18011 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.040/18012
• Subject:
  Euclidean space;
  additive function;
  mid-convex function;
  continuity;
  continuum;
  analytic set;
  Ger-Kuczma classes;
  Primary;
  26B05;
  54D05;
  Secondary;
  26B25;
  54C05;
  54C30
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.040
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other