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Let $C$ be a closed convex subset of a real Hilbert space containing theorigin, and assume that $K$ is the homogenization cone of $C$, i.e., thesmallest closed convex cone containing $C \times \{1\}$. Homogenization conesplay an important role in optimization as they include, for instance, thesecond-order/Lorentz/"ice cream" cone.  In this note, we discuss the polar cone of $K$ as well as an algorithm forfinding the projection onto $K$ provided that the projection onto $C$ isavailable. Various examples illustrate our results.

The Homogenization Cone: Polar Cone and Projection