hm2 - ISyE 7682 Fall 2010 2nd Homework Problem 0.1 Let Ci...

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ISyE 7682 Fall 2010 – 2nd Homework Problem 0.1 Let C i ⊂ ℜ n i be a convex set for i = 1 , 2. Show that aF C 1 × aF C 2 = aF ( C 1 × C 2 ) , ri C 1 × ri C 2 = ri ( C 1 × C 2 ) . Problem 0.2 ±or a nonempty closed convex C ⊆ ℜ n , show that C = { d ∈ ℜ n : C + d C } . Problem 0.3 Let ∅ n = C ⊂ ℜ n be a convex set. Show that: C = ǫ> 0 cl ( 0 <t ǫ tC ) . Problem 0.4 A polyhedron is a set C which can be represented as C = { x ∈ ℜ n : Ax b } for some A ∈ ℜ m × n and b ∈ ℜ n . Assume that the polyhedron C = { x ∈ ℜ n : Ax b } is nonempty and bounded. Show that C = { d ∈ ℜ n : Ad 0 } and that the (possibly empty) polyhedron { x ∈ ℜ n : Ax b } is bounded for any other right hand side b ∈ ℜ m . Problem 0.5 ±or a nonempty closed convex set
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