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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 $d \geq 3$.

Added January 24 2017

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 $n \geq 3$.

Added Further

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 $B$?

I came across the term geodesic in Wikipedia, but I do not know if it is applicable to this question.

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Jose Arnaldo Bebita Dris
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    You need to give more context to your question. Not only the dimension, but also the space (or subspace) itself is relevant to the answer. Anyway, no, straight lines are NOT the shortest objects joining two points. – Edu Jan 24 '17 at 14:58
  • Okay, adding more context to the question now. – Jose Arnaldo Bebita Dris Jan 24 '17 at 14:59
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    What's the definition of a straight line? In the Poincarre disc for example it looks "curvey", hence, Edu's remark is important... – imranfat Jan 24 '17 at 15:00
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    I have quite often seen "straight line" (or just "line") defined as a curve representing the (locally) shortest distance between two points. In other contexts, straight lines are defined as curves whose parametrisation fulfills certain differential equations (a mathematical description of the notion that it "doesn't curve"). Whichever definition is chosen, one should prove the other one as a theorem. – Arthur Jan 24 '17 at 15:01
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    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
  • @Pythagoricus, yes! Does that imply that the shortest distance between any two points in Euclidean $n$-space is a straight line? – Jose Arnaldo Bebita Dris Jan 24 '17 at 15:10
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    Is the shortest distance between two points in Euclidean nn-space always a straight line? First of all distance is a number, not a set of points, so it cannot be a line. So define what the shortest distance means to you. Depending on the definition it might or might not be. – freakish Jan 24 '17 at 15:12
  • @freakish, I guess what I meant to ask is: "Is the length of a particular straight line always the shortest distance between two points in Euclidean $n$-space?" – Jose Arnaldo Bebita Dris Jan 24 '17 at 15:14
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    Well, technically speaking, you have to rigorously define what a ‘straight’ line is in order to get a serious mathematical answer. For practical purposes though, you can imagine that even $\mathbb{R}^4$ or $\mathbb{R}^{18}$ (etc.) have straight lines, and their measure is the shortest distance between their two endpoins. So yes, the shortest distance between two points in a Eucledian space is ‘given by a straight line’. – Pythagoricus Jan 24 '17 at 15:15
  • That is, given two (arbitrary) points $A$ and $B$ in Euclidean $n$-space $\mathbb{R}^n$, is the length of the line segment $\overline{AB}$ the shortest distance between $A$ and $B$? – Jose Arnaldo Bebita Dris Jan 24 '17 at 15:15
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    @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
  • By the way, I'm sorry for the formula mistyped in my previous comment! – Pythagoricus Jan 24 '17 at 15:17
  • @Pythagoricus, that is OK! =) – Jose Arnaldo Bebita Dris Jan 24 '17 at 15:27
  • I think, what you're looking for is examples of spaces where the property in question does not hold, is that so? – Pythagoricus Jan 24 '17 at 15:36
  • Yes, an example of a space which provides a negative answer to my question would do, @Pythagoricus. – Jose Arnaldo Bebita Dris Jan 24 '17 at 15:43
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    Then, have a look at this: https://en.wikipedia.org/wiki/Taxicab_geometry In this geometry we use a slightly different metric (taxicab metric) and the shortest distance between two points is not necessarily given by a straight line! (Of course, we speak intuitively, since ‘staight line’ and ‘distance’ are subject to rigorous definition.) – Pythagoricus Jan 24 '17 at 15:48
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    Alternative metrics for $\mathbb{R}^n$ are a subject of (without being limited to) metric spaces topology. – Pythagoricus Jan 24 '17 at 15:50
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    With euclidean metric, if you ask the shortest distanve formally you are asking over all possible paths ( that are rectifiable, other way it's not possible to measure them) which one of them has smallest lenght, and those are well defined notions with integrals. This paths have the property that are arbitrarily well approximated by any polygonal that has vertices inside your path and that each edge is sufficiently small. So the problem is reduced to look at polygonal paths. – Santropedro Jan 24 '17 at 16:04
  • @Pythagoricus and Santropedro, thank you for those hints. I will take a look at the search engine results returned for those keywords that you have provided. – Jose Arnaldo Bebita Dris Jan 24 '17 at 16:09

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