Exercises
[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.
Show that
\begin{equation} d_ E(x)≥ \max \{ 0, (d_ A(x)-r)\} ~ ~ .\label{eq:d_ E_ d_ A_ r} \end{equation}62Show that in ?? you have equality if \(M=ℝ^ N\).
Give a simple example of a metric space where equality in ?? does not hold.
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]).
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