bounded below by , it achieves a unique minimum on it. Pythagorean inequality. By cosine law, , which must be , since minimizes in , and is convex. Pythagorean equality when is in the relative interior of . If , then since is in the relative interior, we can move from in the direction opposite of , to decrease , contradiction. Thus . Classification theorems The only symmetric Bregman divergences on are squared generalized Euclidean distances (Mahalanobis distance), that is, for some positive definite . The following two characterizations are for divergences on , the set of all probability measures on , with . Define a divergence on as any function of type , such that for all , then: The only divergence on that is both a Bregman divergence and an f-divergence is the Kullback–Leibler divergence. If , then any Bregman divergence on that satisfies the data processing inequality must be the Kullback–Leibler divergence. (In fact, a weaker assumption of "sufficiency" is enough.) Counterexamples exist when . Given a Bregman divergence , its "opposite", defined by , is generally not a Bregman divergence. For example, the Kullback-Leiber divergence is both a Bregman divergence and an f-divergence. Its reverse is also an f-divergence, but by the above characterization, the reverse KL divergence cannot be a Bregman divergence. Examples Squared Euclidean distance is the canonical example of a Bregman distance, generated by the convex function The squared Mahalanobis distance, which is generated by the convex function . This can be thought of as a generalization of the above squared Euclidean distance. The generalized Kullback–Leibler divergence is generated by the negative entropy function When restricted to the simplex, this gives , the usual Kullback–Leibler divergence. The Itakura–Saito distance, is generated by the convex function Generalizing projective duality A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes and vice versa, while preserving incidence and above-below relationships. There are numerous analytical forms of the projective dual: one common form maps the point to the hyperplane . This mapping can be interpreted (identifying the hyperplane with its normal) as the convex conjugate mapping that takes the point p to its dual point , where F defines the d-dimensional paraboloid . If we now replace the paraboloid by an arbitrary convex function, we obtain a different dual mapping that retains the incidence and above-below properties of the standard