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December 16th, 2018, 11:31 AM   #1
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Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Convex Functions

I've ran across two definitions for convex functions. One is taking the second derivative to see if it's positive, and another is some weird definition;


$f$ is called convex if:
${\displaystyle \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).} \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).$

I'm just wondering if these are the same or if they are two different ideas.
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December 16th, 2018, 11:43 AM   #2
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If the function is twice differentiable, testing the second derivative suffices. Otherwise, the "weird" definition (or an equivalent definition) should be used.
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December 16th, 2018, 11:46 AM   #3
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Originally Posted by skipjack View Post
If the function is twice differentiable, testing the second derivative suffices. Otherwise, the "weird" definition (or an equivalent definition) should be used.
Thank you! I guess it's not that weird, it's a good definition. It's just unfamiliar to me!
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December 16th, 2018, 11:57 AM   #4
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Thank you! I guess it's not that weird, it's a good definition. It's just unfamiliar to me!
It helps to draw a picture, since the concept is much simpler than the formal definition.
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December 16th, 2018, 07:33 PM   #5
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Convex function
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December 17th, 2018, 11:35 AM   #6
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Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
Wow! Thank you! Sweet!
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