EDB — 0RC

Exercises

1. [0RC] Topics:fattened set.Prerequisites:[0R9]↺↻.

   Consider a metric space \((M,d)\). Let \(A⊆ M\) be closed and non-empty, let \(r{\gt}0\) be fixed, and let \(d_ A\) be the distance function defined as in eqn. [(9.62)]↺↻. Let then \(E=\{ x,d_ A(x)≤ r\} \), notice that it is closed.

   1. Show that

      \begin{equation} d_ E(x)≥ \max \{ 0, (d_ A(x)-r)\} ~ ~ .\label{eq:d_ E_ d_ A_ r} \end{equation}

      62

   2. Show that in ?? you have equality if \(M=ℝ^ N\).

   3. Give a simple example of a metric space where equality in ?? does not hold.

   4. If \(M=ℝ^ n\), given \(A⊂ ℝ^ n\) closed non-empty, show that \(E=A ⊕ D_ r\) where \(D_ r{\stackrel{.}{=}}\{ x, |x|≤ r\} \) and

      \[ A ⊕ B{\stackrel{.}{=}}\{ x+y,x∈ A, y∈ B\} \]

      is the Minkowski sum of the two sets (see also Section [2CP]↺↻).

   Solution 1

   [0RD]↺↻

   The set \(\{ x,d_ A(x)≤ r\} =A ⊕ D_ r\) is sometimes called the ”fattening” of \(A\). In figure 3 we see an example of a set \(A\) fattened to \(r=1,2\); the set \(A\) is the black polygon (and is filled in), whereas the dashed lines in the drawing are the contours of the fattened sets. ¹ See also the properties in sections [2CP]↺↻ and [2CQ]↺↻.