Norm-Cone Conjugation and Fenchel-Type Duality Beyond Convexity

We introduce a norm-cone conjugation scheme for extended-real-valued functions on normed spaces.

The construction replaces affine minorants by translated norm-cones of the form \(x\mapsto r-\alpha\|x-x_0\|\), with \(\alpha\ge0\), and establishes a nonlinear conjugation framework underlying a Fenchel-type duality theory beyond convexity.

The resulting conjugate is indexed by slopes and centres, and the associated biconjugate is the supremum of all norm-cone minorants lying below the function.

We prove Fenchel--Young type inequalities, introduce admissible slopes and admissible heights, and characterize exact biconjugation in terms of norm-cone supportability.

We also define a norm-cone subdifferential and relate it to exact support and biconjugation.

Finally, we develop an abstract perturbation duality theory based on partial norm-cone conjugation in the perturbation variable.

Weak duality holds without convexity assumptions, while strong duality follows from metric lower-bound conditions, including uniform lower Lipschitz estimates and lower calmness of the value function.