Recall sA : R" → R is a support function given by SA(x) = sup p•x PEA %3D Let A, B be nonempty closed and convex. Show that if A C B with A, BC R" then SA(x) < SB(x) E R" Hint: You can use the fact that for any closed convex set C, we have C and sc are related by C = {p € R" : p •x< sc(x) Vd e R"} %3D Assume A C R", show that sa is monotone.
Recall sA : R" → R is a support function given by SA(x) = sup p•x PEA %3D Let A, B be nonempty closed and convex. Show that if A C B with A, BC R" then SA(x) < SB(x) E R" Hint: You can use the fact that for any closed convex set C, we have C and sc are related by C = {p € R" : p •x< sc(x) Vd e R"} %3D Assume A C R", show that sa is monotone.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Recall sA : R" → R is a support function given by
SA(x) = sup p•x
pEA
(a) Let A, B be nonempty closed and convex. Show that if A C B with A, B C
R" then
SA(x) < SB(x) x € R"
Hint: You can use the fact that for any closed convex set C, we have C and
sc are related by C = {p € R" : p •x< sc(x) Vd E R"}
(b) Assume AC R, show that sa is monotone.
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