Convex variational problems by Bildhauer M.

By Bildhauer M.

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Then (/1 t 12)* = It + g . 1) PROOF. 2). For s E JR, (/1 t h)*(s) = + h(X2)J} = sUPXt+x2=AS(XI +X2) - II(xl) - h(x2)] = sUPXt,xJS(XI +X2) - II (XI) - h(x2)] SUPXt [SXI - II (x])] + SUPX2[SX2 - h(x2)] = and we recognize suPx {sx - infxt + x2 =x[f1 (x]) It(s) + Iz*(s) in this last expression. o The dual version of this result is that, if II and 12 are two closed convex functions finite at some common point, then (/1 + 12)* = It t g . 1, and their conjugates are II and fz respectively; hence (/t t Iz*)* = II + fz .

1, convexity of Ion I = [a, b] implies its upper semi-continuity at a and b. 1). We will therefore content ourselves with checking the convexity of a given function on an open interval. Then, checking convexity on the closure of that interval will reduce to a study of continuity, usually much easier. 1 Let 1 be continuous on an open interval I and possess an increasing right-derivative, or an increasing left-derivative, on I. Then 1 is convex on I. PROOF. Assume that 1 has an increasing right-derivative D+I.

U {+oo} for t f(xo + td) t t f(xo) =: q(t) to, in which case Xo + td i a. It follows 3 Continuity Properties 17 f(xo + td) = f(xo) + tq(t) --+ f(xo) + (xo - a)l =: f(a+) E lR U {+oo} . Then let t t to in the relation f(a) - f(xo) q(t) :,;; q(to) = for all t E ]0, toe a Xo - to obtain l= f(a+) - f(xo) Xo - a :,;; f(a) - f(xo) Xo - a , o hence f(a+) :,;; f(a). The prooffor b uses the same arguments. 3, and is nothing more than the slope-function [f(x) - f(xo)]/(x - xo). 3, to avoid the unpleasant division by x - Xo < O.

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