Let \(E\) be a real Banach space and let \(\overline{B}_{r}(E)\) be the closed ball of radius \(r\) around the origin in \(E\). The \(I\)-conjugate of a bounded below function \(f:\overline{B}_{r}(E)\rightarrow\mathbb{R}\) is defined via the formula \(f^{I}(x^{\ast})=\sup\left\{ x^{\ast}(x)-f(x):x\in\overline{B} _{r}(E)\right\} \) for \(x^{\ast}\in\overline{B}_{r}(E^{\ast}).\) By this restriction of the domain through which \(x\) runs, the \(I\)-conjugate will never take the value \(+\infty\). The paper under review describes the properties of the corresponding Legendre-Fenchel transformation \(f\mapsto f^{I},\) that maps the set \(\mathcal{F}_{r}(E)\) (of all \(r\)-Lipschitz continuous convex functions defined on \(\overline{B}_{r}(E))\) into \(\mathcal{F}_{r}(E^{\ast}).\) This is an involution in the sense that \((f^{I})^{I}|_{\overline{B}_{r}(E)}=f\) for all function \(\mathcal{F}_{r}(E).\) Among the many results proved by the author, we mention here the analogues of the Fenchel duality theorem (Theorem 3.4 in the text) and the duality theorem à la Artstein-Avidan and Milman (Theorem 6.2).