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A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies
Author(s) -
Giannopoulos Apostolos A.
Publication year - 1990
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s002557930001295x
Subject(s) - mathematics , unit sphere , combinatorics , subspace topology , regular polygon , unit (ring theory) , mathematical analysis , geometry , mathematics education
LetA n ( a , b ) = { ( x i ) ∈ R n : ∑ i = 1 n − 1x i 2 ⩽ a 2 , | x n | ⩽ b} .It is proved that for suitable a and b , n ≥7, one can have V n ( A n ) = V n ( B n ) andV n − 1( H ∩ A n ) < V n − 1( H ∩ B n )for every ( n –1)‐dimensional subspace H of ℝ n , where B n is the unit ball of ℝ n . This strengthens previous negative results on a problem of H. Busemann and C. M. Petty.