  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 $D$.
