Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V ) denote the absolute projection constant of V. We show that λ(V ) ≤ λ(Vn) where Vn is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that 4/π = λ(l2 ) ≥ λ(V ) for any two-dimensional real symmetric space V.

Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define λ N n = sup{λ(V) : dim(V) = n, V ⊂ l (N) ∞ }, λn = sup{λ(V) : dim(V) = n}. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ2 = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented.

Abstract. Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, c-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

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