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

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 $K$ in 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 neighborhood of zero in ${\mathbb{R}}^n$). This implies that a mid-convex function $f:D\to \mathbb{R}$, 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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