Is there any simple proof of the following statement: for all vectors $ v,w,u\in V\setminus\{0\} $, where $ V $ is a Euclidean space, inequality $$ \angle(u,v)\le\angle(u,w)+\angle(w,v)$$ holds.
Unfortunately, couldn't find anything useful in books or Google. I've seen this post: Triangle inequality for angles, but I'm not sure if the given answer is correct or not, and is there more clear proof or not.
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That inequality is only true if you are careful about the numerical value assigned to an angle and how you add angles.
The dot product definition gives signed angles.
If you measure angles by the (nonnegative) great circle arclength they cut off on the unit sphere what should happen when the sum wraps around to more than a full circle?
If all you care about are small unsigned angles then you can use that arclength - it's just the triangle inequality for great circle distances.
Yes. First, take the spanning set of the three vectors, allowing us to reduce the problem to 3/D space. We then set the magnitude of $u, v, w$ to 1 as this does not affect angle, and the find these three points on a sphere. The triangle inequality holds for minor arcs on a sphere, and the arc length is equal to the angle, so the required result holds.
Here is a proof of the triangle inequality on spherical surfaces.