On the size of approximately convex sets in normed spaces

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Let X be a normed space. A set A ⊆ X is approximately convex if d(ta + (1 − t)b, A) ≤ 1 for all a, b ∈ A and t ∈ [0, 1]. We prove that every n-dimensional normed space contains approximately convex sets A with H(A,Co(A)) ≥ log2 n− 1 and diam(A) ≤ C √ n(lnn), where H denotes the Hausdorff distance. These estimates are reasonably sharp. For every D > 0, we construct worst possible approximately convex sets in C(0, 1) such that H(A,Co(A)) = diam(A) = D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.