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December 16th, 2018, 10:31 AM   #1
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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, 10:43 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, 10:46 AM   #3
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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.
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, 10:57 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, 06:33 PM   #5
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Convex function
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December 17th, 2018, 10:35 AM   #6
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Wow! Thank you! Sweet!
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