On directional Whitney inequality
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This paper studies a new Whitney type inequality on a compact domain Ω⊂ℝ^d that takes the form inf_Q∈Π_r-1^d(ℰ)f-Q_p ≤ C(p,r,Ω) ω_ℰ^r(f, diam(Ω))_p, r∈ℕ, 0<p≤∞, where ω_ℰ^r(f, t)_p denotes the r-th order directional modulus of smoothness of f∈ L^p(Ω) along a finite set of directions ℰ⊂𝕊^d-1 such that span(ℰ)=ℝ^d, Π_r-1^d(ℰ):={g∈ C(Ω): ω^r_ℰ (g, diam (Ω))_p=0}. We prove that there does not exist a universal finite set of directions ℰ for which this inequality holds on every convex body Ω⊂ℝ^d, but for every connected C^2-domain Ω⊂ℝ^d, one can choose ℰ to be an arbitrary set of d independent directions. We also study the smallest number 𝒩_d(Ω)∈ℕ for which there exists a set of 𝒩_d(Ω) directions ℰ such that span(ℰ)=ℝ^d and the directional Whitney inequality holds on Ω for all r∈ℕ and p>0. It is proved that 𝒩_d(Ω)=d for every connected C^2-domain Ω⊂ℝ^d, for d=2 and every planar convex body Ω⊂ℝ^2, and for d≥ 3 and every almost smooth convex body Ω⊂ℝ^d. [See the pre-print for the complete abstract - not included here due to arXiv limitations.]
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