Discussion Overview
The discussion focuses on the containment property of polar cones for sets in R^n, specifically exploring the relationship between two sets S1 and S2 and their polar cones S1* and S2*. Participants are attempting to prove that if S1 is contained in S2, then S2* is contained in S1*. The conversation also touches on the definition and properties of polar cones, as well as a related challenge involving a specific set defined by a matrix transformation.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant suggests that proving the containment property seems obvious, especially in R^2, but struggles to find a formal proof.
- Another participant questions whether the proof is evident from the definition based on the inner product.
- A participant provides a reasoning based on the inner product, stating that if S1 is contained in S2, the region satisfying the condition for S1 is the same or larger than that for S2, implying S2* is contained in S1*.
- A new challenge is introduced regarding the polar cone of a set defined as S = {x ∈ ℝ^n: x = Ap, p ≥ 0}, prompting further exploration of its polar cone S*.
- Another participant attempts to define S* using the condition x^T a ≤ 0 for all x in S but expresses difficulty in concluding anything about "a" from this definition.
Areas of Agreement / Disagreement
Participants express differing levels of confidence regarding the proof of the containment property, with some finding it straightforward while others seek further clarification. The discussion regarding the polar cone of the set defined by the matrix transformation remains unresolved, with no consensus on the conclusions about "a".
Contextual Notes
The discussion includes assumptions about the properties of inner products and polar cones, and the reasoning presented relies on these definitions. There are unresolved mathematical steps in deriving the polar cone for the specific set defined by the matrix transformation.
