Lower convex envelope

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the lower convex envelope 
  
    
      
        
          
            
              f
              ˘
            
          
        
      
    
    {\displaystyle {\breve {f}}}
  
 of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 defined on an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 is defined at each point of the interval as the <a href="/facts/Supremum/oXCKVopz">supremum</a> of all <a href="/facts/Convex_function/IbzG5SLF">convex functions</a> that lie under that function, i.e.
</p>

f
              ˘
            
          
        
        (
        x
        )
        =
        sup
        {
        g
        (
        x
        )
        ∣
        g
        
           is convex and 
        
        g
        ≤
        f
        
           over 
        
        [
        a
        ,
        b
        ]
        }
        .
      
    
    {\displaystyle {\breve {f}}(x)=\sup\{g(x)\mid g{\text{ is convex and }}g\leq f{\text{ over }}[a,b]\}.}

Lower convex envelope open-in-new

Lower convex envelope