Timeline for Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.

9 events

when toggle format	what		by	license	comment
Apr 13, 2011 at 22:00	history	edited	Chao Li		edited tags
Apr 13, 2011 at 22:00	vote	accept	Chao Li
Apr 13, 2011 at 22:00	answer	added	Chao Li		timeline score: 1
Apr 6, 2011 at 13:05	history	edited	Chao Li		edited tags
Apr 6, 2011 at 0:03	comment	added	Chao Li		Thanks for the hint. In 3 dimensional case, the sphere is a diamond shape and the cross-section parallel to one of this faces is a regular hexagon, which can be proved inside a $\ell_1^2$ sphere. On high dimensional case, it is same as the example given by Yemon Choi. However, I'm not sure how to embed it in $\ell_1^{n-1}$.
Apr 5, 2011 at 23:24	comment	added	David Eppstein		Hint: what is the shape of the $\ell_1^3$ sphere, and what is the shape of a cross-section parallel to one of its faces?
Apr 5, 2011 at 20:00	comment	added	Yemon Choi		Oh, I misread your question. Well, try counting extreme points of ${\mathcal B}_n\cap {\mathcal P}$ when $a_1=\dots=a_n=1$; I think that may lead towards an answer (but I haven't checked the details)
Apr 5, 2011 at 19:50	history	edited	Chao Li	CC BY-SA 2.5	added 2 characters in body; added 2 characters in body
Apr 5, 2011 at 19:08	history	asked	Chao Li	CC BY-SA 2.5