10.4 Distance function[2C4]

E293

[0R9]Topics:distance function.

1. Show that \(d_ A\) is a Lipschitz function.

2. Show that \(d_ A≡ d_{\overline A}\).

3. Show that \(\{ x,d_ A(x)=0\} =\overline A\).

4. If \(M=ℝ^ n\) and \(A\) is closed and non-empty, show that the infimum in ?? is a minimum.

See also 1 and 2. Hidden solution: [UNACCESSIBLE UUID ’0RB’]

E293

[0RC] Topics:fattened set.Prerequisites:1.

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. ??. 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}

   294

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 12.6).

Hidden solution: [UNACCESSIBLE UUID ’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 12.6 and 12.7.