"Irrational" Convexity

Tuesday, August 22, 2017 - 3:30pm
Vitali Milman (University of Tel Aviv)

In the talk we demonstrate many different constructions which are  
analogous to constructions of irrational numbers from rationals. We  
show, following Il.Molchanov, that the solutions of "quadratic"  
equations like   Z^o  =  Z + K always exists (where Z^o is the polar  
body of Z ; Z  and  K  are convex compact bodies containing 0 in the  
interior). Then we show how the geometric mean may be defined for any  
convex compact bodies K  and  T (containing 0 into their interior).
We also construct  K^a for any centrally symmetric K and 0 < a < 1, and also
Log K  for K containing  the euclidean ball D (and K = -K).
  Note,  the power a cannot be above 1 in the definition of power !
  All these constructions may be considered also for the infinite  
dimensional setting, but this is outside the subject of the talk.
  The talk will be well understood by any graduate student.
  These results are joint with Liran Rotem.

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