Walrasian budget set 的 Convexity 問題 - 經濟

※ 引述《washburn (Just a game)》之銘言：
By definition, If X is a convex set, then
: x'' = alpha*x + (1-alpha)*x' belongs to LR+, for any alpha belongs to [0,1].

To show B{p,w} is convex, for any a, b belong to B{p,w},
we need to show that c= alpha*a + (1-alpha)*b also belongs to B{p,w}.

First, c belongs to LR is obvious.
Second, p*c=p*(alpha*a + (1-alpha)*b)
=alpha*p*a+(1-alpha)*p*b
<=alpha*w + (1-alpha)*w
<= w
By definition, B{p,w} is a convex set.
Q.E.D.

: Walrasian budget set：B {p,w} = {x belongs to LR: p*x <= w}。
: 請問，為什麼在 X 為 convex 時，B {p,w} 也是 convex？


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