Algebraic Topology: L5, affine spaces, 9-13-16, part 1

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favorite part of the video so far is panning up to the robot figures sitting on the top shelf

benshapiro
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OK... I'm seven years late, but the bit with the equilateral triangle and the distance to the barycentre (/centroid) confused me for a bit. It's actually two-thirds the *height* of the triangle; not two-thirds the side length. Since the height is slightly less than the side length x (it's sqrt(3)/2 * x), it's not showing that the inequality is sharp as mentioned in the video — to do that you'd have to show that there's a point two-thirds of the *side length* away from the nearest vertex, which is a slightly larger quantity the two-thirds the height and presumably isn't achieved since the vertices seem to achieve the supremum already.

xanderlewis