Minkowski Inequality


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Minkowski Inequality

 

an inequality of the form

where ak and ak (k = 1,2, . . . , n) are non-negative numbers and r > 1. The Minkowski inequality has analogs for infinite series and integrals. The inequality was established by H. Minkowski in 1896 and expresses the fact that in n-dimensional space, where the distance between the points x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) is given by

the sum of the lengths of two sides of a triangle is greater than the length of the third side.

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Varosanec, Properties of mappings related to the Minkowski inequality, Mediterranean J.
The purpose of this paper first is to establish the Minkowski inequality for the dual Quermassintegral sum, which is a generalization of the Minkowski inequality for mixed intersection bodies.
1 The Minkowski inequality for dual Quermassintegral sum of mixed intersection bodies
The following Minkowski inequality for mixed intersection bodies will be established: If K, L [member of] [[phi].
If t [greater than or equal to] 1, then by Minkowski inequality we obtain
It follows easily from Fubini-Tonnelli's theorem forp = 1, the generalized Minkowski inequality (see [4], p.
Therefore, by using the generalized Minkowski inequality we get
By the Holder inequality and the generalized Minkowski inequality, the first term on the right hand side of (3.
Proof From (12), (13), (21) and notice for i < n - 1 to use the Minkowski inequality for integral [14)P.
Proof From (15), (18), (19) and in view of the Minkowski inequality for integral [14)P.
Note first that by the Holder or Minkowski inequality, with [X.
A more general version of the dual Minkowski inequality is: If 0 [less than or equal to] i [less than or equal to] n - 2, then