This from Book "Analysis for Applied Mathematics"- W. E. Cheney, Ex 1.2, Problem 24
Let $<a,b> = \{\lambda a + (1-\lambda)b:0 \leq \lambda \leq 1\}$ be a line segment in a linear space. A polygonal path joining points $a$ and $b$ is any finite union of line segments $\cup_{i=1}^{n}<a_{i},a_{i+1}>$ where $a_{1} = a$ and $a_{n+1} = b$. If the linear space has a norm, the length of the polygonal path is $\Sigma_{i=1}^{n}\|a_{i} - a_{i+1}\|$. Give an example of a pair of points $a,b$ in a normed linear space and a polygonal path joining them such that the polygonal path is not identical to $<a,b>$ but has the same length.
This has stumped me for a long time now. How is it possible without violating the triangle inequality on the norms?
The question, then proceeds as
A path of length $\|a-b\|$ connecting $a$ and $b$ is called a geodesic path. Prove that any geodesic polygonal path connecting $a$ and $b$ is conatined in the set $\{x:\|x-a\|\leq\|b-a\|\}$
I'm more concerned with the example. The second part of the question is for completion's sake.