[Paper Review] Modulus of supporting convexity and supporting smoothness

Grigory Ivanov|arXiv (Cornell University)|Mar 31, 2015

Advanced Banach Space Theory2 references3 citations

TL;DR

This paper introduces the modulus of supporting convexity and supporting smoothness to quantify how the unit sphere of a Banach space deviates from arbitrary supporting hyperplanes. It establishes that the modulus of supporting convexity is equivalent to the classical modulus of convexity at zero, and the modulus of supporting smoothness is equivalent to the modulus of smoothness and Banaáás modulus at zero, proving a Day–Nordlander-type result for these new moduli.

ABSTRACT

We introduce the moduli of the supporting convexity and the supporting smoothness of a Banach space, which characterize the deviation of the unit sphere from an arbitrary supporting hyperplane. We show that the modulus of supporting smoothness, the Bana{ś} modulus, and the modulus of smoothness are all equivalent at zero, the modulus of supporting convexity is equivalent at zero to the modulus of convexity. We prove a Day--Nordlander type result for these moduli.

Motivation & Objective

• To define and analyze new geometric moduli—supporting convexity and supporting smoothness—that characterize the deviation of the unit sphere from arbitrary supporting hyperplanes in a Banach space.
• To establish the equivalence of the modulus of supporting convexity to the classical modulus of convexity at zero, and of the modulus of supporting smoothness to the modulus of smoothness and Banaáás modulus at zero.
• To prove a Day–Nordlander-type estimate for the new moduli, relating them to the Lipschitz constant of the metric projection onto a hyperplane.
• To provide sharp estimates for the modulus of supporting smoothness using inverse functions of the modulus of convexity, particularly in Hilbert spaces.

Proposed method

• Define the modulus of supporting convexity and supporting smoothness based on the geometry of the unit sphere relative to supporting hyperplanes at boundary points.
• Use the Birkhoff-James orthogonality and quasiorthogonality to analyze vector configurations and derive inequalities involving the norm and supporting functionals.
• Employ the standard moduli of convexity $\delta_X(\varepsilon)$ and smoothness $\rho_X(\tau)$ as reference points for equivalence comparisons.
• Apply geometric arguments involving intersecting chords and central symmetry to derive lower bounds for the modulus of supporting smoothness.
• Use the inverse function $\delta_X^{-1}$ to relate the modulus of supporting smoothness to the modulus of convexity, particularly in the Hilbert space case.
• Prove equivalence at zero using the asymptotic comparison $f(t) \asymp g(t)$ as $t \to 0$, defined via positive constants bounding the ratio of functions near zero.

Experimental results

Research questions

• RQ1 How can the deviation of the unit sphere from an arbitrary supporting hyperplane be quantified in a Banach space?
• RQ2 Are the new moduli of supporting convexity and smoothness equivalent at zero to the classical moduli of convexity and smoothness?
• RQ3 Can a Day-Nordlander-type inequality be established for the modulus of supporting smoothness in terms of the Lipschitz constant of the metric projection?
• RQ4 What is the sharp estimate for the modulus of supporting smoothness in terms of the inverse of the modulus of convexity?
• RQ5 Does the equality case in the upper bound for the modulus of supporting smoothness occur in $L_p$ spaces for $p \in (1, \infty)$?

Key findings

• The modulus of supporting convexity is equivalent at zero to the classical modulus of convexity $\delta_X(\varepsilon)$, i.e., $\text{supp}_X(\varepsilon) \asymp \delta_X(\varepsilon)$ as $\varepsilon \to 0$.
• The modulus of supporting smoothness $\lambda_X^+(r)$ satisfies $1 - \frac{1}{2}\delta_X^{-1}(1 - \frac{r}{2}) \leq \lambda_X^+(r)$, with equality approached in Hilbert spaces.
• The modulus of supporting smoothness is equivalent at zero to both the modulus of smoothness $\rho_X(\tau)$ and the Banaáás modulus, i.e., $\lambda_X^+(r) \asymp \rho_X(r) \asymp \text{Ban}_X(r)$ as $r \to 0$.
• A Day-Nordlander-type estimate is proven: $\lambda_X^+(r) \geq \lambda_X^-(r) \geq \lambda_X^-(1) \cdot \delta_X^{-1}(1 - \frac{r}{\xi_X})$, with $\xi_X$ being the norm of the supporting functional.
• In Hilbert spaces, the upper bound $\lambda_X^+(r) \leq \delta_X(2r)$ is achieved, confirming sharpness of the estimate.
• The conjecture is proposed that the upper bound in the estimate $\lambda_X^+(r) \leq \lambda_X^-(1) \cdot \delta_X^{-1}(1 - \frac{r}{\xi_X})$ becomes an equality for $L_p$ spaces with $p \in (1, \infty)$.