Abstract:
In the paper, our main aim is to introduce a new concept and call it the
$L_p$-mixed
geominimal surface area$G_{p}(K_{1},\,\dots\,,K_{n})$
of
$n$
convex bodies
$K_{1},\,\dots\,,K_{n}$,
which obeys the classical basic properties.
The new affine geometric
quantity in a special case yields Petty's geominimal surface area
$G(K)$
of a convex
body
$K$,
Lutwak's
$p$-geominimal surface area
$G_{p}(K)$
of
$K$,
and the newly established
mixed geominimal surface area
$G(K_{1},\,\dots\,,K_{n})$
of
$n$
convex bodies
$K_{1},\,\dots\,,K_{n}$.
We establish some
$L_{p}$-mixed geominimal surface area inequalities
for the
$L_p$-mixed geominimal surface area, whose some special cases are Petty's
geominimal surface area inequality, Lutwak's
$p$-geominimal surface area inequality, and
some new mixed geominimal surface area inequalities.
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