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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 \(D\).

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
Full Text: DOI arXiv Euclid
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