Let C be a nonempty closed convex subset of X and let p = C and...

 

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Let C be a nonempty closed convex subset of X and let p = C and let x = X. Recall that pPc(x) if and only the obtuse angle condition (Ve EC) (c-p, x-p) <0 holds. Also recall that a vector g belongs to Nc(p), the normal cone of C at p, if (Vc C) (g,c-p) 0. Use this to prove that No = Pc - Id, where P is the set-valued inverse of Pc. Indeed, you only need to check that Nc(p)=-p+P(p). (This looks much harder than it is!) Let C be a nonempty closed convex subset of X and let p = C and let x = X. Recall that pPc(x) if and only the obtuse angle condition (Ve EC) (c-p, x-p) <0 holds. Also recall that a vector g belongs to Nc(p), the normal cone of C at p, if (Vc C) (g,c-p) 0. Use this to prove that No = Pc - Id, where P is the set-valued inverse of Pc. Indeed, you only need to check that Nc(p)=-p+P(p). (This looks much harder than it is!)

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To prove that No P textId where Pdagger is the setvalued inverse of Pc and Ncp represents the normal ... View the full answer