The identity operator on an n-dimensional space E is an ''exposed'' point of the unit ball of the space L(E) of all bounded operators on E. More precisely, the identity operator \(I_ E\) is the only operator u such that tr u\(=n\) and \(\| u\| \leq 1\) [cf. \textit{D. Garling}, Proc. Cambridge Philos. Soc. 76, 413-414 (1974; Zbl 0286.47020)]. The present paper gives estimates for the ''exposing modulus'' \(\delta\) (t) which is the best function with the property that \(\| u\| \leq 1\) and tr \(u\geq n(1-\delta (t))\) imply \(\| I_ E-u\| \leq t\). The results are as follows. (Only real spaces are considered here). We have \(t^ 2/2n^ 2\leq \delta (t)\leq t\) for any E. As an application, the authors prove the following inequality relating the 1-absolutely summing norm \(\pi_ 1(E)\) of the identity on E and the norm \(\gamma_{\infty}(E)\) of factorization of \(I_ E\) through \(L_{\infty}:\) \[ \gamma_{\infty}(E)\{1-[n^ 2-\pi_ 1(E)^ 2]^{1/2}\}\leq n^{-1}\pi_ 1(E). \] The authors also study the spaces E for which \(\delta\) (t) is asymptotically as large as possible, i.e. there is a constant \(c>0\) such that \(\delta\) (t)\(\geq ct\) for all t. In that case they say that \(I_ E\) is sharply exposed. They show that this holds if the group of isometries of E is finite. The proof uses the theory of numerical ranges. Finally, the authors consider the particular cases of spaces with an unconditional (or symmetric) basis with constant 1. In the unconditional case, they show that for some constant \(c>0\) either \(\delta\) (t)\(\geq ct\) for all t or \(\delta (t)\leq ct^ 2\) for all t. In the symmetric case, they show that either \(E=\ell^ n_ 2\) isometrically, or \(I_ E\) is sharply exposed.