  Relative smoothness and strong convexity have recently gained considerable
attention in optimization. These notions are generalizations of the classical
Euclidean notions of smoothness and strong convexity that are known to be dual
to each other. However, conjugate dualities for non-Euclidean relative
smoothness and strong convexity remain an open problem as noted earlier by Lu,
Freund and Nesterov [SIAM J. Optim., 28 (2018), pp. 333-354]. In this paper we
address this question by introducing the notions of anisotropic strong
convexity and smoothness as the respective dual counterparts. The dualities are
developed under the light of generalized conjugacy which leads us embed the
anticipated dual notions within the superclasses of certain upper and lower
envelopes. In contrast to the Euclidean case these inclusions are proper in
general as showcased by means of counterexamples.
