Talk:Radial set

Every radial set is a star domain although not conversely.

This seems wrong given the current definition. I think the definition needs to be stronger for that. I think one needs something like

(I) ∃a₀∈A ∀x∈X ∃tₓ>0 : {a₀ + t⋅x∣t∈[0,tₓ]} = {a₀+t⋅x∣t≥0}∩A

instead of just

(II) ∃a₀∈A ∀x∈X ∃tₓ>0 : {a₀ + t⋅x∣t∈[0,tₓ]}⊆A

Otherwise, consider a U-shape. This is clearly not a star domain. But given a point in the upper left corner, it appears that (II) is satisfied.

n-lab also gives a different definition: https://ncatlab.org/nlab/show/radial+set

One can augment this definition by saying that A is radial around a₀∈A, iff r⋅(A-a₀) ⊆ A-a₀ for all r∈[0,1]. This should be equivalent to (I) and not (II). Rslz (talk) 19:03, 24 October 2023 (UTC)

To a layperson, this also seems like a false statement. My argument is simply that the definition of star domain given by the article always permits adding arbitrary points to A. Tbardwell (talk) 17:55, 24 June 2026 (UTC)