# Is the shortest distance between two points in Euclidean $n$-space always a straight line?

This question is related to these two questions [1] and [2].

Here is my question:

Is the shortest distance between two points in Euclidean $n$-space always a straight line?

In two dimensions, the answer is YES (?). I am unsure how to answer the same question for the case of dimension $d \geq 3$.

For the case of Euclidean $n$-space (i.e., $\mathbb{R}^n$), the shortest distance between two points is indeed a straight line for the case $n=2$, a result which is obtained by considering the distance formula (which is an example of what we call a metric) for the length of the line segment $\overline{{P_1}{P_2}}$:

$$d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where the point $P_1$ has coordinates $(x_1, y_1)$ and the point $P_2$ has coordinates $(x_2, y_2)$.

I am unsure of how to answer my question for the case of Euclidean $n$-spaces, for dimension $n \geq 3$.

That is, given any two arbitrary points $A, B \in {\mathbb{R}}^n$, is the length of the line segment $\overline{AB}$ the shortest distance between $A$ and $B$?
• You know that the Eucledian metric in $\mathbb{R}^n$ is $d(P_a,P_b) = \sqrt{\sum_{i=1}^{\infty} (a_i - b_i)^2}$, don't you? – Pythagoricus Jan 24 '17 at 15:09
• @JoseArnaldoBebitaDris Ok, so now we get into another trouble. What is the length of a line? Or a path? You see, if you define it via integrals, then yes. The shortest path between two points $A, B$ is a line segment between them. Can be defined as convex hull for example. – freakish Jan 24 '17 at 15:17