Math Problem Solver

— Let 𝑃 = {𝑝0, . . . , 𝑝𝑛} ⊂ R𝑑 be a finite set of points. We now compare

VR𝑟(𝑃) for different values of 𝑟.

(a) Determine a value 𝑟0 such that VR𝑟(𝑃) has only 0-simplices if and only if 𝑟 < 𝑟0.

(b) Determine a value 𝑟max such that VR𝑟(𝑃) is the full 𝑛-simplex on 𝑃 if and only if

𝑟max ≤ 𝑟.

(c) Show that there are finitely many jump-values for the Vietoris-Rips complex,

that is, there are finitely many values 𝑟0 < 𝑟1 < · · · < 𝑟𝑚 = 𝑟max such that

VR𝑟(𝑃) = VR𝑟𝑖

(𝑃) if and only if 𝑟𝑖 ≤ 𝑟 < 𝑟𝑖+1. What is the maximal number of

jump values for a set of 𝑛 points?

(d) Give a set of four points with as few as possible respectively as many as possible

jump values.