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A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line segment between two points on the graph lies above the graph, the region above the graph is convex, etc. I want to know why the word "convex" goes with the inequality in this direction, and how I can remember it. Every reason I have heard makes just as much sense applied to the opposite inequality ("concave down").

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    You are compromising the utility of the notion by failing to relativize it to subintervals of the domain. About 80% of the time what you are interested in in Calc 1 regarding concavity is where the the sign of the concavity (equivalently, the sign of the curvature) changes, ie, in any INFLECTION POINTS.2011-09-29
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    [The epigraph of a convex $\mathbb{R} \to \mathbb{R}$ function is a convex set](http://t.co/ktO1PVOV8Z) within ``graph space'' $(x,y)$ although this doesn't explain why we should look at the epigraph instead of the subgraph….2014-06-19
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    I think sometimes as mathematicians we forget that some words we use do have normal "every-day" meanings. The definition of convex, in the every-day sense, means that a surface bulges out TOWARDS you as you look at it. And since the canonical orientations of the way we draw graphs have us think we are standing BELOW the graph, a convex function looks... well... convex.2017-03-10

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