Timeline for Generalization of Borsuk-Ulam to arbitrary ratio

6 events

when toggle format	what		by	license	comment
Sep 7, 2015 at 19:55	vote	accept	Erel Segal-Halevi
May 24, 2015 at 21:12	comment	added	Real		Ah yes, something like that. The idea was to make the lines $f_1=k$ and $f_2=k$ parallel, by making both $f_1$ and $f_2$ a function only of the distance to two faces as you describe, or more simply a positive function of $x_m \cdot x$ (where $\cdot$ is the dot product). Indeed I think something like this gives a generalization, and a way to built $g$ such that $g(x)=(y,...,y) /iff x=x_m, x \cdot x_m = 0 or x=-x_m$. To that end, just choose two positive functions s.t. $f_1=h_1(x_m \dot x)$, $f_1=h_1(x_m \dot x)$, $h_1(y)=-h_1(y)$,$h_1(1)=-h(-1)=1=h_2(1)=-h_2(-1)$ that only intersect at 0,1,-1.
May 24, 2015 at 19:04	comment	added	Erel Segal-Halevi		Maybe you mean something like this: tube.geogebra.org/student/m1236259 Suppose f is 1 on the green face and -1 on the red face, and its value elsewhere depends only on its distance from the green face (e.g. it has a single value on the entire yellow line). Apparently there can be many functions that satisfy this condition, and they don't have to coincide anywhere in the blue region.
May 24, 2015 at 18:48	comment	added	Erel Segal-Halevi		What do you mean by the "*" symbol?
May 22, 2015 at 15:12	history	edited	Real	CC BY-SA 3.0	added 8 characters in body
May 22, 2015 at 14:53	history	answered	Real	CC BY-SA 3.0