Minkowski thickening

Core idea

For $X\subseteq\mathbb R^d$ and $r>0$ , the Minkowski thickening of $X$ by radius $r$ is the Minkowski sum

X+B_r=\{x+b:x\in X, |b|<r\}.

Equivalently, it is the open $r$ -neighborhood $\{z:\operatorname{dist}(z,X)<r\}$ . Using the closed ball produces the closed thickening and changes only boundary conventions.

Scaling and translation

For $s>0$ , $s(X+B_r)=sX+B_{sr}$ . Translation commutes with thickening. These identities make the construction natural in multiscale estimates.

Effect on porosity

If $X$ is $\nu$ -porous on lines and $0<\nu'<\nu$ , then $X+B_r$ remains $\nu'$ -porous on segments of length larger than $r/(\nu-\nu')$ , as long as the original upper scale is respected.

References

Rolf Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014. DOI record.