Discussion Overview
The discussion revolves around finding the normal cone of a closed convex subset in a Hilbert space, specifically focusing on the unit disc in a two-dimensional Euclidean space. Participants explore the implications of different points within the set and how they affect the normal cone.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant defines the normal cone \( N_{K}(x_{0}) \) for a closed convex subset \( K \) in a Hilbert space and poses a specific case with \( K \) as the unit disc.
- Another participant asserts that if \( x_{0} = (0,0) \), then the normal cone \( N_{K}((0,0)) \) consists of just one point, \( (0,0) \).
- A later reply confirms that \( x_{0} = (0,0) \) is indeed the point of interest.
- Another participant questions the normal cone when \( x_{0} = (0,1) \) and asks for its value.
- One participant suggests that the normal cone \( N_{K}((0,1)) \) includes all points of the form \( (0,c) \) where \( c \geq 0 \), though they express uncertainty about this claim.
Areas of Agreement / Disagreement
Participants express differing views on the normal cone at various points within the set, indicating that multiple competing interpretations exist regarding the normal cone's definition and its specific values at given points.
Contextual Notes
There is some uncertainty regarding the completeness of the arguments presented, particularly concerning the normal cone at \( x_{0} = (0,1) \) and the conditions under which these claims hold.