December 16th, 2018, 11:31 AM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 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: ${\displaystyle \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1t)x_{2})\leq tf(x_{1})+(1t)f(x_{2}).} \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1t)x_{2})\leq tf(x_{1})+(1t)f(x_{2}).$ I'm just wondering if these are the same or if they are two different ideas. 
December 16th, 2018, 11:43 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,105 Thanks: 1907 
If the function is twice differentiable, testing the second derivative suffices. Otherwise, the "weird" definition (or an equivalent definition) should be used.

December 16th, 2018, 11:46 AM  #3 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  
December 16th, 2018, 11:57 AM  #4 
Senior Member Joined: Aug 2012 Posts: 2,135 Thanks: 621  
December 16th, 2018, 07:33 PM  #5 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,750 Thanks: 614 Math Focus: Yet to find out.  
December 17th, 2018, 11:35 AM  #6 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research! 
Wow! Thank you! Sweet!


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