= Face duality for polyhedral cones
{title2=$\tau\longleftrightarrow C^\vee\cap\tau^\perp$}
For a closed polyhedral cone $C$, the maps $\tau\mapsto C^\vee\cap\tau^\perp$ and $\nu\mapsto C\cap\nu^\perp$ give inverse order-reversing bijections on faces. Choose a exposing functional in the relative interior of the dual face. The bipolar identity and an exposing functional for the original face show that applying the maps twice returns that face.
// Target: geometry-and-topology.bigb
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