Normal structure and Pythagorean approach in Banach spaces

Summary

Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>X$ be a real Banach space and $S(X) = \{x \in X: \|x\| = 1\}$ be the unit sphere of $X$. The parameters $E_{\epsilon}(X)=\sup\{\alpha_{\epsilon}(x): x \in S(X)\}$, $e_{\epsilon}(X)=\inf\{\alpha_{\epsilon}(x): x \in S(X)\}$, $F_{\epsilon}(X)=\sup\{\beta_{\epsilon}(x): x \in S(X)\}$, and $f_{\epsilon}(X)=\inf\{\beta_{\epsilon}(x): x \in S(X)\}$, where $\alpha_{\epsilon}(x) = \sup\{\| x + \epsilon y \|^{2}+ \| x - \epsilon y \|^{2}: y \in S(X)\}$ and $\beta_{\epsilon}(x) = \inf\{\| x + \epsilon y \|^{2}+ \| x - \epsilon y \|^{2}: y \in S(X)\}$, are defined and studied. The main result is that a Banach space $X$ with $E_{\epsilon}(X) < 2 + 2\epsilon +\frac{1}{2}\epsilon^{2}$ for some $0\leq \epsilon \leq 1$ has uniform normal structure.