He Homework # 4 Problem 1 : Power Cone
نویسنده
چکیده
• Kα is nonempty, because |0| ≤ 01 · · · 0m = 0 ⇒ 0 ∈ Kα. • Kα is closed w.r.t. addition, because for any given (x, y), (x, y) ∈ Kα, we have |y + y| ≤ |y|+ |y| (by triangle inequality) ≤ (x1) · · · (xm) + (x1) · · · (xm) ≤ (x1 + x1) · · · (xm + xm) . • Kα is closed w.r.t. multiplication, because for any given (x, y) ∈ Kα, and any λ ∈ R+, we have |λy| = λ|y| ≤ λ(x1 1 · · ·xm m ) = λα1+···+αm(x α1 1 · · ·xm m ) = (λx1) · · · (λxm) ⇒ (λx, λy) ∈ Kα.
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