By (31), the Minkowski inequality
, and the generalized Holder inequality we have
From the "Minkowski inequality
" (Proposition 4.6) we can obtain analogous estimates if we take the 2 in the last position and move it backwards making it take every position from the last to the first; in other words, considering the following exponents:
By the Holder inequality, Minkowski inequality
combined with the estimate of (39), and Young inequality, we obtain
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.
If t [greater than or equal to] 1, then by Minkowski inequality
It follows easily from Fubini-Tonnelli's theorem forp = 1, the generalized Minkowski inequality
(see , p.
The new Orlicz-Minkowski and Brunn-Minkowski inequalities for the Orlicz mean dual affine quermassintegrals in special case yield the [L.sub.p]-dual Minkowski inequality
and Brunn-Minkowski inequalities for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and Brunn-Minkowski inequalities for general volumes.
Using the Minkowski inequality
([4, p.21]) for r > 1 and followed by the Holder inequality ([4, p.22]) with exponents s - t/r - t > 1 and s - t/s - r, we obtain
Proof From (12), (13), (21) and notice for i < n - 1 to use the Minkowski inequality
for integral [14)P.147], we obtain for p [greater than or equal to] 1
Note first that by the Holder or Minkowski inequality
, with [X.sub.m] and [Y.sub.m] defined by (2.2),
A Minkowski inequality
for mixed quermassintegrals states that, for K,L [member of] [K.sup.n] and 0 [less than or equal to] i < n -1,