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

TL;DR

The paper establishes conditions under which additive and convex functions are continuous based on their boundedness on non-flat continua or non-meager analytic subsets in b^n, linking geometric properties to functional regularity.

Contribution

It proves that sums of non-flat continua have non-empty interior and that boundedness on such sets ensures the continuity of mid-convex functions, connecting geometric and functional analysis.

Findings

Sum of n copies of a non-flat continuum has non-empty interior.

Boundedness on non-flat continua implies continuity of mid-convex functions.

Sum of n copies of a non-meager analytic subset is not meager and has interior.

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 $D$.