In this paper we introduce the multiple-description ℓ1-compression problem: minimize kz1k1+λkz2k1 subject to the three distortion constraints kA1z1 −yk2 ≤ δ1, kA2z2 −yk2 ≤ δ2, and k1/2(A1z1 + A2z2) − yk2 ≤ γ. This problem formulation is interesting in, e.g., ad-hoc networks where packets can be lost. If a description (z2) is lost in the network and only one description is received (z1), it is still possible to decode and obtain a signal quality equal or better than described by the parameter δ1 (and vice versa). If both descriptions are received, the quality is determined by the parameter γ. This problem is difficult to solve using first-order projection methods due to the intersecting second-order cones. However, we show that by recasting the problem into its dual form, one can avoid the difficulties due to conflicting fidelity constraints. We then show that efficient first-order ℓ1-compression methods are applicable, which makes it possible to solve large scale problems, e.g., multiple-description ℓ1-compression of video.