Determining the metric space recall

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1. For any set F and point x in a metric space recall that d(x, F ) := inf{d(x, y): y ∈ F }. Let F be a closed set in a normed linear space S with the usual distance d(x, y) := x - y . Show that d(·, F ) isa convex function if and only if F is a convex set.

2. Let U be a convex open set in Rk . For any set A ⊂ Rk let - A := {-x : x ∈ A}. Assume U = -U , so that 0 ∈ U . Let µ be a measure defined on the Borel subsets of U with 0 <>(U ) <>∞ and µ(B) ≡ µ(-B). Let f be a convex function on U . Show that f (0) ≤ ( f dµ/µ(U ). Hint: Use the image measure theorem 4.2.8 with T (x ) ≡ -x.

Reference no: EM13990823

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