p***c
发帖数: 2403 | 1 我有一个非常简单的分段函数，定义在[0，\infty)上，
a>0, c>0，d>0 是三个常数
f(x)=a if x
f(x)=-d if x>=c
现在要求这样一个g函数：
g(x)=sup{ v(x) | v is convex, and v(y)\leq f(y) for all y\geq 0}
简单的说，g是所有小于f的凸函数的上确界
从图上直观我的来说 g 就应该是一条直线把(0,a)和(c,-d)连起来，在x>c的地方不变
怎么具体证明啊？ 谢谢 | l********e
发帖数: 3632 | 2 geometrically convex function has upper lever set convex. Hence your convex
set is the intersection of all convex set with your properties. which is
also called convex-hull. Use the definition of convex-hull you can see
easily that your answer is correct.
You can also prove algebraically which is even more trivial, since you
already has one convex function constructed, then if the sup is not your
function, a easy contradiction can be derived from the fact if some point in
(0, a) has larger value than your linear function.
【在 p***c 的大作中提到】
: 我有一个非常简单的分段函数，定义在[0，\infty)上，
: a>0, c>0，d>0 是三个常数
: f(x)=a if x: f(x)=-d if x>=c
: 现在要求这样一个g函数：
: g(x)=sup{ v(x) | v is convex, and v(y)\leq f(y) for all y\geq 0}
: 简单的说，g是所有小于f的凸函数的上确界
: 从图上直观我的来说 g 就应该是一条直线把(0,a)和(c,-d)连起来，在x>c的地方不变
: 怎么具体证明啊？ 谢谢
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