- E10
[0B0] Fix \(I=\{ 1,\ldots n\} \). Let \(n\) distinct points \(y_ 1,\ldots y_ nββ\) be given; let \(π:Iβ I\) be a permutation for which triangle inequalities between successive points are equalities i.e.
\[ |y_{π(i+2)}-y_{π(i+1)}| + |y_{π(i+1)}-y_{π(i)} | =|y_{π(i+2)}-y_{π(i)} | \]for \(i=1,\ldots n-2\). Show that there are only two, we call them \(π_ 1,π_ 2\). Tip: Show that any such permutation necessarily puts the points βin orderβ, i.e. you have
\[ β i,y_{π_ 1(i+1)}{\gt} y_{π_ 1(i)}\quad ,\quad β i,y_{π_ 2(i+1)}{\lt} y_{π_ 2(i)} \](up to deciding which is \(π_ 1\) and which is \(π_ 2\)).
Solution 1
EDB β 0B0
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English
Authors:
"Mennucci , Andrea C. G."
.
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