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    STUDIA MATHEMATICA - Issue no. 1 / 2022  
         
  Article:   CHARACTERIZATIONS OF HILBERTIAN NORMS INVOLVING THE AREAS OF TRIANGLES IN A SMOOTH SPACE.

Authors:  TEODOR PRECUPANU.
 
       
         
  Abstract:  
DOI: 10.24193/subbmath.2022.1.10

Published Online: 2022-03-10
Published Print: 2022-03-31
pp. 145-149

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In an earlier paper, we have defined the heights of a nontrivial triangle with respect to Birchoff orthogonality in a real smooth space $X$, $mbox{dim}, mbox{X} geq 2$. In the present paper, we remark that, generally, the area of a nontrivial triangle have not the same value for different heights of the triangle. The proposal of this paper is to characterize the norms of the space having property that the area of any triangle is well defined (independent of considered height). In this line, we give six equivalent properties using the directional derivative of the norm. For example, the area is well defined for all triangles if and only if Birchoff orthogonality is symmetric. Consequently, according to a well known result of Leduc, if $mbox{dim} Xgeq 3$ each of those six properties characterizes the hilbertian norms (generated by inner products).

Keywords: Smooth space, strictly convex space, norm derivative, Birkho_ orthogonality, height of a triangle, hilbertian norm.

Mathematics Subject Classification (2010): 54AXX.
 
         
     
         
         
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