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In recent years, a new area of mathematics — idempotent or "tropical" mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky.
The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space.
The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated