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We are told that a function is convex only if the following inequality holds:

f(tx + (1 - t) x' , ty + (1 - t) y') ≤ tf(x, y) + (1-t) f(x', y')

for 0 ≤ t ≤ 1 and all pairs of points (x, y) and (x', y').

What does this mean geometrically?


marked as duplicate by Namaste, A. Goodier, mlc, Raskolnikov, Ethan Bolker Mar 28 '18 at 21:34

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Draw the graph of a convex function and pick two points on the graph. The line interpolating these two points lies above the graph.


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