The continuity of additive and convex functions which are upper bounded on non-flat continua in R^n - Banakh, Taras; Jabłońska, Eliza; Jabłoński, Wojciech (Topological Methods in Nonlinear Analysis, 2019)
We prove that for a continuum KR^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 AK 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 neighbourhood of zero in R^n). This implies that a mid-convex function fD→R defined on an open convex subset DR^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.