The so-called Minkowski difference of convex bodies (compact and convex
subsets of R
n) can be seen as the subtraction counterpart of the Minkowski or
vectorial addition (of convex bodies in R
n). If K, E are convex bodies, then the
Minkowski difference of K and E is defined as
K ~ E := {x ? R
n
: x + E ? K},
i.e., the largest set to which we can (Minkowski) add E and the sum remains
contained in K. For a convex body K, the convex body E is a (Minkowski)
summand of K if there exists another convex body L such that E +L = K. We
notice, that (K ~ E) + E ? K, and this content may be strict.
The (relative) inradius r(K; E) of K with respect to E is defined as
r(K; E) = sup{r : ? x ? R
n with x + r E ? K}.
For 0 = ? = r(K; E), the inner parallel body of K (relative to E) at distance ?
is the Minkowski difference of K and ?E:
K ~ ?E = {x ? R
n
: x + ?E ? K}.
We write K? to denote the (relative) inner and outer parallel bodies of K,
i.e., the so-called full system of (relative) parallel bodies of K:
K? := (
K ~ |?|E for - r(K; E) = ? = 0,
K + ?E for 0 = ? < 8.
When ? = 0, it clearly coincides with K, whereas for ? = -r(K; E), we obtain
the kernel of K, relative to E. For convex bodies K, E, it is known that the
kernel does not have interior points. Futher, the full system of parallel bodies of
K, relative to E, is concave with respect to inclusion and Minkowski addition.
Let K, E be convex bodies, and let K have interior points. Then, the isoperimetric quotient is the ratio
I(K; E) = S(K; E)
n
vol (K)
n-1
where vol (K) denotes the volume of K and S(K; E) is the relative surface area
of K (with respect to E). When E is the Euclidean ball, the latter coincides
with the classical isoperimetric quotient.
The monotonicity of the isoperimetric quotient of the family of inner (and
outer) parallel bodies of a convex body has received some new attention recently,
as it happens, for example, to be connected to the Eikonal abrasion model.
We will analyze several aspects of the family of inner parallel bodies of a
convex body, in particular, several results about parallel bodies within the realm
of the Brunn-Minkowski Theory, based on the concavity of the full system of
parallel bodies. Our aim is to prove that the isoperimetric quotient is decreasing
in the parameter of definition of parallel bodies, along with a characterization
of those convex bodies for which that quotient happens to be constant on some
interval within its domain. The result will be obtained relative to arbitrary
gauge bodies, having the classical Euclidean setting as a particular case.
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