Proving local minimal to be global minimal in Convex optimization

Problem:

min f(x) . s.t. g(x)<= 0 and h(x) = 0

f(x) and g(x) are convex, h(x) is affine

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Analysis

assume x is local optimal value.
f(x) <= f(y) for all feasible y, || x - y ||<= p

we need to prove f(x) <= f(y) for all feasible y

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Prove: