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1. ### Equidistant curve of an ellipse

How do you define "equidistant"? Here's the definition I assume you are using: If C is a curve, and E is an ellipse, then C is "equidistant from E" if there is some positive real number r such that for all x in C, inf{|x-y| : y in E} = r. This is actually an interesting question. There is a...
2. ### Lebesgue measurable sets

FAIL. From wikipedia: http://en.wikipedia.org/wiki/Outer_measure#Formal_definitions Defining properties of an outer measure: * The empty set has measure 0. * Monotonicity: If A is a subset of B, then the measure of A is at most the measure of B. * Countable Subadditivity: The measure of a...
3. ### Fitting Spheres into Arbitrary Geometry

If your polytope is convex, then your job isn't too hard. Assume first that the origin lies in the interior of the polytope. Let F1, ..., Fn be faces of the polytope. Let u1, ..., un be unit normals to the faces, and t1, ... tn be the distance of each face from the origin. Let r be the radius...
4. ### Topological spaces and basis

Suppose a U is an open set in R2 (consider the usual euclidean topology - we are drawing a distinction between R2 and RxR). Then for each point x in U, we can find an open disc Dx about x small enough so that it fits in U. If we like, we can also find a "rectangle" Rx small enough to fit inside...
5. ### Projection of a differentiable manifold onto a plane

It sounds like you'll be embedding this 3-manifold into R4. If this is the case, it's more a matter of linear algebra than anything else. You can approximate this manifold as a union 3-simplexes (pyramids with four triangular sides) in R4. Perhaps you'll have 1000 of these simplexes to form a...
6. ### 3rd grade geometry question

The trick is that words can often be left out in English, but are still part of the implied meaning. "the number of faces that share an edge with any one face" actually means: "the number of faces that share an edge with any one particular face". Your interpretation of the meaning of the...
7. ### Smallest Set Containing All n-point Sets

Yes, but that's not the smallest. :wink: Take the intersection of the balls about x0 and y0. This would also necessarily contain P. About the "translations and orthogonal transformations" business: I'm looking for a closed form for a set S which must "be able to contain" P. By "be able to...
8. ### Smallest Set Containing All n-point Sets

I've been thinking about this. Suppose you have an n-point set P in Rm which has the property that for any two points x, y in P, ||x - y|| < 2. If we fix n, what can we say about the smallest set S in Rm that contains P, allowing for both translations and orthogonal transformations of S? If...
9. ### Definition of *straight* lines

I think eok20 hit the nail on the head with this: This notion of "straightness" relies only on a prior notion of "length". So, if you can assign a real number to each one-dimensional connected set of points, then you have enough to define straightness.