We prove that for a continuum K ⊂ R n the sum K +n of n copies of K has non-empty interior in R n if and only if K is not flat in the sense that the affine hull of K coincides with 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 ⊂ K the sum A +n of n copies of A is not meager in R n (and then the sum A +2n of 2n copies of the analytic set A has non-empty interior in R n and the set (A − A) +n is a neighborhood of zero in R n ). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ 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.1991 Mathematics Subject Classification. 26B05, 26B25, 54C05, 54C30, 54D05.

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