Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality”
T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ).
Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form
Tf =
⎧ , f’ ≥ 0,
⎨
⎩ , f’ < 0,
with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions...
Our main intention in this paper is to demonstrate how some seemingly purely geometric notions can be presented and understood in an analytic language of inequalities and then, with this understanding, can be defined for classes of functions and reveal new and hidden structures in these classes. One main example which we discovered is a new duality transform for convex
non-negative functions on attaining the value 0 at the origin (which we call “geometric convex
functions”). This transform, together...
In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space X₁ = (ℝⁿ,||·||₁) with the property that if a space X₂ = (ℝⁿ,||·||₂) is "not too far" from X₁ then there exists a [λn]-dimensional subspace E⊂ ℝⁿ such that E₁ = (E,||·||₁) and...
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