PACS. 82.70.Uv -Surfactants, micellar solutions, vesicles, lamellae, amphiphilic systems (hydrophilic and hydrophobic interactions). PACS. 68.18.Jk -Surface and interfaces; thin films and low-dimensional systems (structures and nonelectric properties): Phase transitions. PACS. 61.30.St -Lyotropic phases.Abstract. -We demonstrate the possibility of a stable equilibrium multi-lamellar ("onion") phase in pure lamellar systems (no excess solvent) due to a sufficiently negative Gaussian curvature modulus. The onion phase is stabilized by non-linear elastic moduli coupled to a polydisperse size distribution (Apollonian packing) to allow space-filling without appreciable elastic distortion. This model is compared to experiments on copolymer-decorated lamellar surfactant systems, with reasonable qualitative agreement.Introduction. -The existence of vesicles at thermal equilibrium is a long-standing, and still controversial, problem [1]. Unilamellar and multi-lamellar vesicles (MLVs) were first induced in lyotropic lamellar phases with excess water by adding energy (e.g., shear flow, ultrasound, electric field) [2]. In a very elegant work, using membranes generated by a chemical reaction, Hoffman and co-workers demonstrated that any of lamellae, unilamellar vesicles, or MLVs can be prepared in the same system, depending on the mechanical path chosen [3]. However, in some special cases [4,5] equilibrium unilamellar vesicles have been demonstrated. These systems are all in the dilute regime (large excess of water). Some experiments have suggested that MLVs (or onions) can be stabilized in the semi-dilute regime (excess water) [5]. Theories to explain the stability of dilute unilamellar vesicles either describe a competition between the entropy of mixing and the curvature energy of the vesicles [5][6][7] or a symmetrybreaking instability leading to a spontaneous curvature [8]. Onions are also predicted to be stabilized in the dilute and semi-dilute regimes due to an unstable curvature energy; in these cases, a transition towards unilamellar vesicles is avoided by imposing either a core energy [9] or a cutoff in the entropy of the Helfrich interactions [10]. Indeed, Simons and Cates [10] have discussed the stability of unilamellar vesicles and the transition to onions as the concentration increases even when the curvature energy is unfavorable, due to entropic reasons.