Ekeland has written several books on popular science, in which he has explained parts of dynamical systems, chaos theory, and probability theory. These books were first written in French and then translated into English and other languages, where they received praise for their mathematical accuracy as well as their value as literature and as entertainment.
Ivar Ekeland applied the Shapley-Folkman lemma to explain Claude Lemarechal's success with Lagrangian relaxation on non-convex minimization problems. This lemma concerns the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowski sum of the four non-convex sets (right) is the sum of four points (+) from the (left-hand) sets--two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown in red).
Ekeland explained the success of methods of convex minimization on large problems that appeared to be non-convex. In many optimization problems, the objective function f are separable, that is, the sum of many summand-functions each with its own argument:
For example, problems of linear optimization are separable. For a separable problem, we consider an optimal solution
with the minimum value For a separable problem, we consider an optimal solution (xmin, f(xmin))
to the "convexified problem", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the limit of a sequence of points in the convexified problem
 An application of the Shapley-Folkman lemma represents the given optimal-point as a sum of points in the graphs of the original summands and of a small number of convexified summands.
This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimizationmethods on problems that were known to be non-convex. Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions. The Shapley-Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.
Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. 28. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). ISBN978-0-89871-450-0. MR1727362. (Corrected reprinting of the 1976 North-Holland (MR463993) ed.)
Ekeland, Ivar (1990). Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. 19. Berlin: Springer-Verlag. pp. x+247. ISBN978-3-540-50613-3. MR1051888.
Aubin, Jean-Pierre; Ekeland, Ivar (2006). Applied nonlinear analysis. Mineola, NY: Dover Publications, Inc. pp. x+518. ISBN978-0-486-45324-8. MR2303896. (Reprint of the 1984 Wiley (MR749753) ed.)
^Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. 28 (Corrected reprinting of the (1976) North-Holland ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. 357-373. ISBN978-0-89871-450-0. MR1727362.
^Aubin, Jean-Pierre; Ekeland, Ivar (2006). Applied nonlinear analysis (Reprint of the 1984 Wiley ed.). Mineola, NY: Dover Publications, Inc. pp. x+518. ISBN978-0-486-45324-8. MR2303896.
^ abKirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN978-0-521-38289-2.
^ abcd(Ekeland 1999, pp. 357-359) harv error: no target: CITEREFEkeland1999 (help): Published in the first English edition of 1976, Ekeland's appendix proves the Shapley-Folkman lemma, also acknowledging Lemaréchal's experiments on page 373.
the inclusion can be strict even for two convex closed summand-sets, according to Rockafellar (1997, pp. 49 and 75) harvtxt error: no target: CITEREFRockafellar1997 (help). Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergent sequences.
Hiriart-Urruty & Lemaréchal (1993, pp. 143-145, 151, 153, and 156): Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1993). "XII Abstract duality for practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 306. Berlin: Springer-Verlag. pp. 136-193 (and bibliographical comments on pp. 334-335). ISBN978-3-540-56852-0. MR1295240.
^ abEkeland, Ivar (1974). "Une estimationa priori en programmation non convexe". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B (in French). 279: 149-151. ISSN0151-0509. MR0395844.
^Aubin (2007, pp. 458-476): Aubin, Jean-Pierre (2007). "14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley-Folkman theorem, pages 463-465)". Mathematical methods of game and economic theory (Reprint with new preface of 1982 North-Holland revised English ed.). Mineola, NY: Dover Publications, Inc. pp. xxxii+616. ISBN978-0-486-46265-3. MR2449499.