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/sci/ - Science & Math

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1893894 No.1893894 [Reply] [Original]

For which metric functions M will the set of all points p in a two dimensional space satisfying M(p) = x, where x is some constant real, be a closed curve?

(forgive me if I'm using the terminology a little loosely, but I'm sure you get my meaning)

>> No.1893912

bump

>> No.1893925

what /?

>> No.1893932

>>1893925
Given that a circle is all of the points a given distance from a center point, under which definitions of "distance" are circles closed curves?

>> No.1893997

I doubt there's any nice criterion for this.

If the metric is continuous then a "circle" around a point will at least include a closed curve, but it might contain a bunch of extra stuff as well. (e.g. imagine defining a metric on the plane by mapping it to a surface shaped like a red blood cell.)

I wonder if it's possible to have a discontinuous metric whose circles are always closed curves...

>> No.1894033

>>1893997
How exactly is continuity defined for two-dimensional functions?

>> No.1894162

The metric function is continuous. if you're using a half decent topology, ie <span class="math">{x}[/spoiler] is closed, then your subset is closed.

>> No.1894813

>>1894033
by defining a topology

>> No.1895025

If there are 360 degrees in a circle how many [unit] are there in a sphere?

>> No.1895792

OP, do you mean M(p,0)=x?
there are a lot.
Taxi cab metric gives you a diamond
"regular" metric gives you the circle.
many more that I'm not going to list.
You can define your own metrics.
Maybe you're asking what the basis of metric functions that have this property is? I don't know how to answer that, but maybe someone else can.

>> No.1895802

I think OP means norm, not metci.

>> No.1895808

>>1895802
metci=metric, sorry