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October 4th, 2011, 02:45 PM   #1
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Math Focus: Topology
Path Connected Set?

Hi all!

This is an old comp problem: how can I show that, in , the set is path connected?
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October 4th, 2011, 03:11 PM   #2
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Re: Path Connected Set?

You can show that the set is convex.
So you have to show:
, whenever and
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October 4th, 2011, 05:24 PM   #3
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Re: Path Connected Set?

Quote:
Originally Posted by guynamedluis
You can show that the set is convex.
So you have to show:
, whenever and
Since the set is points on the surface of the unit sphere in n dimensions, it is not convex.
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October 4th, 2011, 05:56 PM   #4
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Re: Path Connected Set?

Quote:
Originally Posted by mathman
Since the set is points on the surface of the unit sphere in n dimensions, it is not convex.
Oh right.
We are looking at , not .

Then what can you do?
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October 5th, 2011, 01:20 PM   #5
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Re: Path Connected Set?

I presume each point can be expressed in the n dimensional equivalent of polar coordinates, i.e. radius (r) and n-1 angles. Take any two points (r=1) and for each angle (one at a time) get the curve by changing the angle for the first point to the angle for the second point. You will a set of "arcs" on the surface of the hypersphere connecting the first point to the second.
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