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The continuity of additive and convex functions which are upper bounded on non-flat continua in \(\mathbb R^n\). (English) Zbl 1428.26022
Summary: 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 \(K\) is 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 neighbourhood of zero in \(\mathbb R^n)\). This implies that a mid-convex function \(f\colon D\to\mathbb R\) 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 \(D\).

MSC:
26B05 Continuity and differentiation questions
54D05 Connected and locally connected spaces (general aspects)
26B25 Convexity of real functions of several variables, generalizations
54C05 Continuous maps
54C30 Real-valued functions in general topology
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