We answer a question which remained open for some years by proving that a Banach space X can be a semi-L-summand and not an L-summand in its bidual \(X^{**}\). This means that each element y in \(X^{**}\) has a unique best approximation \(\pi\) (y) in X, the metric projection \(\pi\) satisfies that \(\| y\| =\| \pi (y)\| +\| y-\pi (y)\|\) for all y in \(X^{**}\), but \(\pi\) is not linear. Our examples are spaces A(K) of real-valued continuous affine functions on a compact convex set K which is some set of constant width in the dual of a Banach space which is an L-summand in its bidual. In the simplest case K is a product of infinitely many copies of the triangle. The proof involves some nice ideas on infinite-dimensional sets of constant width as well as the notion of pseudoball and the intersection property introduced by P. Harmand and E. Behrends to deal with proper M-ideals. We also prove a nonlinear version of E. Behrends’ L-M theorem.