Minkowski's theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Minkowski%27s_theorem


00:01:21 1 Formulation
00:02:10 2 Example
00:02:29 3 Proof
00:03:09 4 Applications
00:03:18 5 See also
00:07:13 6 References
00:07:57 See also



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SUMMARY
=======
In mathematics, Minkowski's theorem is the statement that any convex set in 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
   which is symmetric with respect to the origin and which has volume greater than 
  
    
      
        
          2
          
            n
          
        
      
    
    {\displaystyle 2^{n}}
   contains a non-zero integer point. The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice 
  
    
      
        L
      
    
    {\displaystyle L}
   and to any symmetric convex set with volume greater than 
  
    
      
        
          2
          
            n
          
        
        
        d
        (
        L
        )
      
    
    {\displaystyle 2^{n}\,d(L)}
  , where 
  
    
      
        d
        (
        L
        )
      
    
    {\displaystyle d(L)}
   denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).