New journal article from Brilliant Light Power

  • If you follow THH's counsel, you will not need to explain non-trivial results such as this one. I would not be surprised if your problem description would become much more manageable and compact if you were to follow his advice.

    Eric I still think I motivated why I would like to include the definition as is, but i also see a value to add what I think THHuxleynew's is indicating. I'm still not convinced that his suggestion can precisely lead to the generalisation I'm looking for to include as well in as suplementary question.

  • No that's not correct. Take the set [a,b] then for each x,y in [a,b] a point p in between x,y also is located in [a,b] hence adding middle numbers does not incrase the set. However if you take the union of [1,2] and [4,5] you can of cause create a larger set

    [1,2]u{3}u[4,5] so you are right sometimes but we consider a measure on all great circles which so this means that all great circkes are included in this set. A measure is define of it's oppinion on measurable sets, what is measurable is defined by the borell

    algebra e.g. basically if you take finite unions and intersection and complements between of all open sets a set is open if can be generated by a countable union of the base set of all sets D{p,r} = all x so that distance(x,p) < r on the sphere which is the intersection of all open sets on E^3 and S^2. So technically measures are not defined on all possible sets, that's the technical difficulty in measure theory and makes precise arguments. A set D{p,r} that is not the empty set has the property that for all two

    normals in that set all normals between those will be in that set so you can't find a normal between those two that is outside D{p,r} that's how I define between. But for the set of all great circles is those with a normal pointing to the unit sphere S2 so if you find a great circle between two others it too has to have a normal pointing to S^2 which is already included in the set of all great circles. Now with a reasonable notion of between there is sets where indeed you can find a loop between, but then whats the issue with that, it has no essential drawbacks for my conjecture.


    Unfortunately the underlying structure here is diffeomorphic to E2, 2D real numbers.


    Therefore any Borel algebra derived from great circles will be deficient. You can use a measure, but it needs to be isomorphic to that induced by the E3 metric on the great circle unit normals as they form an E3 embedding. It must therefore be defined on dense subsets of great circles with uncountable numbers of elements.


    Yet another reason why you might want to follow my suggestion.


    Regards, THH

  • Eric I still think I motivated why I would like to include the definition as is, but i also see a value to add what I think THHuxleynew's is indicating. I'm still not convinced that his suggestion can precisely lead to the generalisation I'm looking for to include as well in as suplementary question.


    Just imagine: a Mathematics Stack Exchange question consisting of two short paragraphs and no bold text or headings. Several upvotes. Two well-reasoned responses. This is the future if we try.

  • So what do you like my added retake of the problem ?

  • From the Wikipedia article: "Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even." Can you specify exactly what you believe ƒ(u) to be in this case, for x a unit vector, and u ∈ C(x) a vector from the set of unit vectors perpendicular to x? I can guess what it might be, but I'll probably guess it incorrectly. Is your function ƒ odd or even? We can take a look at it when you write it down.


    In your alternative formulation, by fiat you have restricted the unit normals to the great circles to be ones that point towards the upper hemisphere. That is less interesting, for it makes a result of 1/2 suspiciously unsurprising.

  • From the Wikipedia article: "Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even." Can you specify exactly what you believe ƒ(u) to be in this case, for x a unit vector, and u ∈ C(x) a vector from the set of unit vectors perpendicular to x? I can guess what it might be, but I'll probably guess it incorrectly. Is your function ƒ odd or even? We can take a look at it when you write it down.


    In your alternative formulation, by fiat you have restricted the unit normals to the great circles to be ones that point towards the upper hemisphere. That is less interesting, for it makes a result of 1/2 suspiciously unsurprising.

    f is an assumed density of mu, every function on the sphere can be written as an even blus and odd function f(p) = (f(p) + f(-p))/2 + ((f(p) - f(-p))/2, then a even id a(-p)=p and odd if a(-p)=-a(p). The uniform density is even.

    and the result of the treansform shall be the uniform density e.g. Y_0^0

  • From the Wikipedia article: "Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even." Can you specify exactly what you believe ƒ(u) to be in this case, for x a unit vector, and u ∈ C(x) a vector from the set of unit vectors perpendicular to x? I can guess what it might be, but I'll probably guess it incorrectly. Is your function ƒ odd or even? We can take a look at it when you write it down.


    In your alternative formulation, by fiat you have restricted the unit normals to the great circles to be ones that point towards the upper hemisphere. That is less interesting, for it makes a result of 1/2 suspiciously unsurprising.

    Yes I should nuke the point to the upper part, that's the optimal version

  • Stefan, can you provide a specific ƒ, rather than making a generic argument that could apply to any ƒ? What is your ƒ(u), for u a vector, in concrete terms? Once you write down a specific function we can examine whether it is even or odd (or neither). Your general argument doesn't lead where you want it to lead, for example, if the even part is 0.


    A density function would presumably be everywhere nonnegative, and so not nontrivially odd. But I doubt your ƒ(u) consists of only a density.