4 I. Convex Sets at Large c∗) Assume that one cannot find a non-zero vector c ∈ Rd and a number α such that c, vi = α for i = 1, . . . , m. Prove that H is injective. Remark: The map H is an example of a moment map see Chapter 4 of [F93]. 4∗ (The Brickman Theorem). Let q1,q2 : Rn −→ R be quadratic forms and let Sn−1 = x ∈ Rn : x = 1 be the unit sphere. Consider the map T : Rn −→ R2, T (x) = ( q1(x),q2(x) ) . Prove that the image T (Sn−1) of the sphere is a convex set in R2, provided n 2. Remark: We prove this in Chapter II (see Theorem II.14.1). 5∗ (The Schur-Horn Theorem). For an n × n real symmetric matrix A = (αij), let diag(A) = (α11, . . . , αnn) be the diagonal of A, considered as a vector from Rn. Let us fix real numbers λ1,... , λn. Consider the set X ⊂ Rn of all diagonals of n × n real symmetric matrices with the eigenvalues λ1,... , λn. Prove that X is a convex set. Furthermore, let l = (λ1, . . . , λn) be the vector of eigenvalues, so l ∈ Rn. For a permutation σ of the set {1,... , n}, let lσ = (λσ(1), . . . , λσ(n)) be the vector with the permuted coordinates. Prove that X = conv lσ : σ ranges over all n! permutations of the set {1,... , n} . Remark: See Theorem II.6.2. 6 (The Birkhoff - von Neumann Theorem). For a permutation σ of the set {1,... , n}, let us define the n × n permutation matrix Xσ = (ξσ ij ) as ξσ ij = 1 if σ(j) = i, 0 if σ(j) = i. Prove that the convex hull of all n! permutation matrices Xσ is the set of all n × n doubly stochastic matrices, that is, matrices X = (ξij), where n i=1 ξij = 1 for all j, n j=1 ξij = 1 for all i and ξij ≥ 0 for all i, j. We consider an n × n matrix X as a point in Rn2. Remark: We prove this in Chapter II (see Theorem II.5.2). 7. Let us fix an even number n = 2m and let us interpret Rn+1 as the space of all polynomials p(τ) = α0 + α1τ + . . . + αnτ n of degree at most n in one variable τ. Let K = p ∈ Rn+1 : p(τ) ≥ 0 for all τ ∈ R be the set of all non-negative polynomials. Prove that K is a convex set and that K is the set of all polynomials that are representable as sums of squares of polynomials of degree at most m: K = k i=1 qi 2 , where deg qi ≤ m .

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