The structure of polarity is examined while providing a glimpse into some of the features of n‐dimensional geometry. The optimization problem associated with minimizing the volume of a dual body is described and acts as a unifying theme for this material. First, we give a gentle (and quick) introduction to aspects of duality, including the notion of polar bodies, and their application to affine invariant points. A discussion of the difficulties associated with volume calculations in high dimensions is also given. Following this, we construct the polar of a general convex polytope and derive expressions for the polar volume of the simplex and cube.