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I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance: https://math.stackexchange.com/a/14469/454016 Poincaré duality is explained through a duality of ...

A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...