A Note on Carlier Inequality
Regina S. Burachik, J. E. Martínez-Legaz
TL;DR
This work extends Carlier's quantitative Fitzpatrick inequality from Hilbert spaces to reflexive Banach spaces by replacing the identity with a strongly monotone operator $B$ that possesses a finite-valued Fitzpatrick function. In the Hilbert setting, taking $B=I$ recovers Carlier's inequality and yields a finite-valued Fitzpatrick representation $F_I(x,v)$; the framework further specializes to $2$-uniformly convex spaces via $B=J_X$, the normalized duality mapping, to obtain a Carlier-type inequality with $F_T(x,v)-<x,v>\ge \frac{\mu}{2\lambda}\sup_{x'\in J_X(x)} ||x-(J_X+\lambda T)^{-1}(x'+\lambda v)||^2$. Additionally, the paper derives an improved Hilbert-space version of the strong Fitzpatrick inequality: $F_T(x,v)-<x,v>\ge \frac{1}{2}\inf_{(w,z)\in G(T)} (||x-w||^2+||v-z||^2)$, which strengthens the known $1/4$-factor bound via the same method with $B=I$ and $\lambda=1$.
Abstract
Recently, Carlier established in [3] a quantitave version of the Fitzpatrick inequality in a Hilbert space. We extend this result by Carlier to the framework of reflexive Banach spaces. In the Hilbert space setting, we obtain an improved version of the strong Fitzpatrick inequality due to Voisei and Zălinescu.