DOI: 10.68381/jca21039 ISSN: 0944-6532
Refinements of the Brunn-Minkowski Inequality
María A. Hernández Cifre, Jesús Yepes Nicolás
The Brunn-Minkowski theorem says that
\mathrm{vol}\bigl((1-\lambda)K+\lambda L\bigr)^{1/n}
v
o
l
(
(
1
−
λ
)
K
+
λ
L
)
1
/
n
, for
K,L
K
,
L
convex bodies, is a concave function in
\lambda
λ
, and assuming a common hyperplane projection of
K
K
and
L
L
, it was proved that the volume itself is concave. In this paper we study refinements of Brunn-Minkowski inequality, in the sense of ‘enhancing’ the exponent, either when a common projection onto an (
n-k
n
−
k
)-plane is assumed or for particular families of sets. In the first case, we show that the expected result of concavity for the
k
k
-th root of the volume is not true, although other Brunn-Minkowski type inequalities can be obtained under the (
n-k
n
−
k
)-projection hypothesis. In the second case, we show that for
p
p
-tangential bodies, the exponent in Brunn-Minkowski inequality can be replaced by
1/p
1
/
p
.