Monotone and convex restrictions of continuous functions

Buczolich, Zoltán (2017) Monotone and convex restrictions of continuous functions. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 452 (1). pp. 552-567. ISSN 0022-247X

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Abstract

Suppose that f belongs to a suitably defined complete metric space C-alpha of Milder alpha-functions defined on [0,1]. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets A subset of [0,1] such that f vertical bar(A) is monotone, or convex/concave. Some of our results are about generic functions in C-alpha like the following one: we prove that for a generic f is an element of C-1(alpha)[0, 1], 0 < alpha < 2 for any A subset of [0,1] such that f vertical bar(A) is convex, or concave we have dim(H) A <= dim(M) A <= max{0, alpha - 1}. On the other hand we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for 1 < alpha <= 2 for any f is an element of C-alpha [0,1] there is always a set A subset of [0,1] such that dim(H) A = alpha - 1 and f vertical bar(A) is convex, or concave on A.

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