Legendre-Fenchel duality in m-convexity

Wibowo, Ratno Bagus Edy
Marjono
Pramana, Eko Dedi

Hilbert Journal of Mathematical Analysis
Vol. 2 No. 2 (2024): Hilbert J. Math. Anal.

The Legendre-Fenchel transform, which maps a function to its convex conjugate, provides a dual perspective that is fundamental in understanding optimization problems. In this work, we show that m-convex function in n-dimensional normed spaces can be viewed as the Legendre-Fenchel dual problem. By constructing an epi-graph of m-convexity in n-dimensional normed spaces, we obtained some properties including the Legendre transform. Particularly, we prove this for certain convex functions.