By Murnaghan F. D.

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We assume that the entries of Φ satisfy for all i, j ∈ {1, . . 11) |Φij | √ d where K > 0 is an absolute constant. We denote by Φ1 , . . , Φd the row vectors of Φ and we define for all p ∈ {1, . . , d} the semi-norm · ∞,p , for x ∈ Rd , by x d ∞,p = max | Φj , x |. 1 j p 1} denote its unit ball. If E = span{Φ1 , . . , Φp } Let B∞,p = {x ∈ R : x ∞,p and PE is the orthogonal projection on E, then B∞,p = PE B∞,p + E ⊥ , moreover, PE B∞,p is a parallelepiped in E. In the next theorem, we obtain an upper bound of the logarithm of the covering numbers of the unit ball of d1 , denoted by B1d , by a multiple of B∞,p .

This means that we need a uniform control of the smallest and largest singular values of all block matrices of A with 2p columns. 3 this is a sufficient condition for the exact reconstruction of m-sparse vectors by 1 -minimization with m ∼ p. When |Ax|2 satisfies good concentration properties, the restricted isometry property is more adapted. In this situation, γ2p ∼ 1. 2). Similarly, an estimate of rad (ker A ∩ B1N ) gives an estimate of the size of sparsity of vectors which can be reconstructed by 1 -minimization.

XN ). 13. — The geometry of faces of A(B1N ). Let 1 ≤ m ≤ n ≤ N . Let A be an n × N matrix with columns X1 , . . , XN ∈ Rn . 2) if and only if one has ∀I ⊂ [N ], 1 ≤ |I| ≤ m, ∀(εi ) ∈ {−1, 1}I , conv({εi Xi : i ∈ I}) ∩ conv({θj Xj : j ∈ / I, θj = ±1}) = ∅. 3) 48 CHAPTER 2. COMPRESSED SENSING AND GELFAND WIDTHS Proof. — Let I ⊂ [N ], 1 ≤ |I| ≤ m and (εi ) ∈ {−1, 1}I . Observe that c y ∈ conv({θj Xj : j ∈ / I, θj = ±1}) iff there exists (λj )j∈I c ∈ [−1, 1]I such that |λj | 1 and y = j∈I c λj Xj .