Modulus and characteristic of convexity

<h2 id="definitions">Definitions</h2>
The modulus of convexity of a Banach space (X, ||⋅||) is the function δ : [0, 2] → [0, 1] defined by

δ
        (
        ε
        )
        =
        inf
        
          {
          
            1
            −
            
              ‖
              
                
                  
                    x
                    +
                    y
                  
                  2
                
              
              ‖
            
            
            :
            
            x
            ,
            y
            ∈
            S
            ,
            ‖
            x
            −
            y
            ‖
            ≥
            ε
          
          }
        
        ,
      
    
    {\displaystyle \delta (\varepsilon )=\inf \left\{1-\left\|{\frac {x+y}{2}}\right\|\,:\,x,y\in S,\|x-y\|\geq \varepsilon \right\},}

where S denotes the unit sphere of (X, || ||). In the definition of δ(ε), one can as well take the infimum over all vectors x, y in X such that ǁxǁ, ǁyǁ ≤ 1 and ǁx − yǁ ≥ ε.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
The characteristic of convexity of the space (X, || ||) is the number ε0 defined by

ε
          
            0
          
        
        =
        sup
        {
        ε
        
        :
        
        δ
        (
        ε
        )
        =
        0
        }
        .
      
    
    {\displaystyle \varepsilon _{0}=\sup\{\varepsilon \,:\,\delta (\varepsilon )=0\}.}

These notions are implicit in the general study of uniform convexity by J. A. Clarkson (Clarkson (1936); this is the same paper containing the statements of <a href="/facts/Clarkson%2527s_inequalities/Gdfx3b7I">Clarkson's inequalities</a>). The term "modulus of convexity" appears to be due to M. M. Day.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="properties">Properties</h2>
<ul><li>The modulus of convexity, δ(ε), is a <a href="/facts/Monotonic_function/MjQK0YL2">non-decreasing</a> function of ε, and the quotient δ(ε) / ε is also non-decreasing on (0, 2].<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The modulus of convexity need not itself be a <a href="/facts/Convex_function/IbzG5SLF">convex function</a> of ε.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> However, the modulus of convexity is equivalent to a convex function in the following sense:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> there exists a convex function δ1(ε) such that</li></ul>

δ
        (
        ε
        
          /
        
        2
        )
        ≤
        
          δ
          
            1
          
        
        (
        ε
        )
        ≤
        δ
        (
        ε
        )
        ,
        
        ε
        ∈
        [
        0
        ,
        2
        ]
        .
      
    
    {\displaystyle \delta (\varepsilon /2)\leq \delta _{1}(\varepsilon )\leq \delta (\varepsilon ),\quad \varepsilon \in [0,2].}

<ul><li>The normed space (X, ǁ ⋅ ǁ) is <a href="/facts/Uniformly_convex_space/H9gGXdy7">uniformly convex</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> its characteristic of convexity ε0 is equal to 0, i.e., if and only if δ(ε) > 0 for every ε > 0.</li>
<li>The Banach space (X, ǁ ⋅ ǁ) is a <a href="/facts/Strictly_convex_space/qX0RMJtV">strictly convex space</a> (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only <a href="/facts/Antipodal_point/2V3228jA">antipodal points</a> (of the form x and y = −x) of the unit sphere can have distance equal to 2.</li>
<li>When X is uniformly convex, it admits an equivalent norm with power type modulus of convexity.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> Namely, there exists q ≥ 2 and a constant c > 0 such that</li></ul>

δ
        (
        ε
        )
        ≥
        c
        
        
          ε
          
            q
          
        
        ,
        
        ε
        ∈
        [
        0
        ,
        2
        ]
        .
      
    
    {\displaystyle \delta (\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].}

<h2 id="modulus-of-convexity-of-the-lpspaces">Modulus of convexity of the LP spaces</h2>
The modulus of convexity is known for the LP spaces.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> If 
 
 
 
 1

(
            
              1
              −
              
                δ
                
                  p
                
              
              (
              ε
              )
              +
              
                
                  ε
                  2
                
              
            
            )
          
          
            p
          
        
        +
        
          
            (
            
              1
              −
              
                δ
                
                  p
                
              
              (
              ε
              )
              −
              
                
                  ε
                  2
                
              
            
            )
          
          
            p
          
        
        =
        2.
      
    
    {\displaystyle \left(1-\delta _{p}(\varepsilon )+{\frac {\varepsilon }{2}}\right)^{p}+\left(1-\delta _{p}(\varepsilon )-{\frac {\varepsilon }{2}}\right)^{p}=2.}

Knowing that 
 
 
 
 
 δ
 
 p
 
 
 (
 ε
 +
 )
 =
 0
 ,
 
 
 {\displaystyle \delta _{p}(\varepsilon +)=0,}
 
 one can suppose that 
 
 
 
 
 δ
 
 p
 
 
 (
 ε
 )
 =
 
 a
 
 0
 
 
 ε
 +
 
 a
 
 1
 
 
 
 ε
 
 2
 
 
 +
 ⋯
 
 
 {\displaystyle \delta _{p}(\varepsilon )=a_{0}\varepsilon +a_{1}\varepsilon ^{2}+\cdots }
 
. Substituting this into the above, and expanding the left-hand-side as a <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> around 
 
 
 
 ε
 =
 0
 
 
 {\displaystyle \varepsilon =0}
 
, one can calculate the 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 coefficients:

δ
          
            p
          
        
        (
        ε
        )
        =
        
          
            
              p
              −
              1
            
            8
          
        
        
          ε
          
            2
          
        
        +
        
          
            1
            384
          
        
        (
        3
        −
        10
        p
        +
        9
        
          p
          
            2
          
        
        −
        2
        
          p
          
            3
          
        
        )
        
          ε
          
            4
          
        
        +
        ⋯
        .
      
    
    {\displaystyle \delta _{p}(\varepsilon )={\frac {p-1}{8}}\varepsilon ^{2}+{\frac {1}{384}}(3-10p+9p^{2}-2p^{3})\varepsilon ^{4}+\cdots .}

For 
 
 
 
 2

δ
          
            p
          
        
        (
        ε
        )
        =
        1
        −
        
          
            (
            
              1
              −
              
                
                  (
                  
                    
                      ε
                      2
                    
                  
                  )
                
                
                  p
                
              
            
            )
          
          
            
              1
              p
            
          
        
        .
      
    
    {\displaystyle \delta _{p}(\varepsilon )=1-\left(1-\left({\frac {\varepsilon }{2}}\right)^{p}\right)^{\frac {1}{p}}.}

Therefore, 
 
 
 
 
 δ
 
 p
 
 
 (
 ε
 )
 =
 
 
 1
 
 p
 
 2
 
 p
 
 
 
 
 
 
 ε
 
 p
 
 
 +
 ⋯
 
 
 {\displaystyle \delta _{p}(\varepsilon )={\frac {1}{p2^{p}}}\varepsilon ^{p}+\cdots }
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Uniformly_smooth_space/715KDNln">Uniformly smooth space</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-86416-4. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0889253">0889253</a>.</li>
<li>Clarkson, James (1936), "Uniformly convex spaces", Transactions of the American Mathematical Society, 40 (3), American Mathematical Society: 396–414, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1989630">10.2307/1989630</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1989630">1989630</a></li>
<li>Fuster, Enrique Llorens. Some moduli and constants related to metric fixed point theory. Handbook of metric fixed point theory, 133–175, Kluwer Acad. Publ., Dordrecht, 2001. <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1904276">1904276</a></li>
<li><a href="/facts/Joram_Lindenstrauss/LW2BeVTd">Lindenstrauss, Joram</a> and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.</li>
<li><a href="/facts/Joram_Lindenstrauss/LW2BeVTd">Lindenstrauss, Joram</a>; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-08888-1.</li>
<li><a href="/facts/Vitali_Milman/4aHLsgu4">Vitali D. Milman</a>. Geometric theory of Banach spaces II. Geometry of the unit sphere. Uspechi Mat. Nauk, vol. 26, no. 6, 73–149, 1971; Russian Math. Surveys, v. 26 6, 80–159.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">p. 60 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Day, Mahlon (1944), "Uniform convexity in factor and conjugate spaces", Annals of Mathematics, 2, 45 (2): 375–385, doi:10.2307/1969275, JSTOR 1969275 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Lemma 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">see Remarks, p. 67 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979). - Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">see Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel Journal of Mathematics, 20 (3–4): 326–350, doi:10.1007/BF02760337, MR 0394135, S2CID 120947324
. <a href="/wiki/Gilles_Pisier" target="_blank">/wiki/Gilles_Pisier</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Hanner, Olof (1955), "On the uniform convexity of 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 and 
 
 
 
 
 ℓ
 
 p
 
 
 
 
 {\displaystyle \ell ^{p}}
 
", Arkiv för Matematik, 3: 239–244, doi:10.1007/BF02589410 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Modulus and characteristic of convexity open-in-new

Modulus and characteristic of convexity