Universität Bremen

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.