This is my problem with mathematics.  The concept of a "soft maximum," the principles behind calculating it, and its startling similarity to a hard maximum, are all fascinating and exciting to me.  Look at that!  Two completely separate functions, and such magical results.  According to this post, it's useful for "convex optimization."  I clicked through to the "related post," which was merely a comment about someone else's opinions on "convex optimization," so I looked it up on Wikipedia:<p><a href="http://en.wikipedia.org/wiki/Convex_optimization" rel="nofollow">http://en.wikipedia.org/wiki/Convex_optimization</a><p>Ah!  A technique used to "minimize convex functions."  Maybe some of this notation will make sense to me if I understand the underlying concept of whatever a "convex function" is.<p><a href="http://en.wikipedia.org/wiki/Convex_functions" rel="nofollow">http://en.wikipedia.org/wiki/Convex_functions</a><p>Great. An entire article that is completely and utterly meaningless to me.  I mean, absolutely nothing in that article--oh!  "Convex sets?"  That looks promising.<p><a href="http://en.wikipedia.org/wiki/Convex_set" rel="nofollow">http://en.wikipedia.org/wiki/Convex_set</a><p>Jackpot!  The pretty pictures make the idea of a convex set clear to me.  Unfortunately, by now I'm 4 clicks away, and my actual understanding of the subject is clearly just scratching the surface.  Connecting my newfound--and obviously still naive--understanding of convexity* to "soft maximums," which initially inspired this search, feels dumbfoundingly impossible.<p>Am I approaching this all wrong?  Am I expecting too much?  Thinking too little?  I would love to understand more about this subject, and I have tried to learn it the same way I learned how to program: by Googling and working on my own problems.  However, the resources simply don't seem to be there in the same way.  What's the deal, here?<p>* Would it be more accurate to say "Euclidian convexity?" What would that mean, exactly?