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 Mar31 awarded Tumbleweed Mar24 asked Does every boundary segment of a convex polyhedron lies on one of its faces? Mar24 comment Are these equivalent definitions of faces of convex sets? @timhortons Thanks a lot. Mar23 revised Are these equivalent definitions of faces of convex sets? added 3 characters in body Mar23 revised Are these equivalent definitions of faces of convex sets? added 2 characters in body Mar23 asked Are these equivalent definitions of faces of convex sets? Mar23 asked Are these inconsistent definitions of extreme subset of convex set? Oct3 comment Will continuous extension preserve strict convexity? This function is not even convex. You can consider $g(x)=h(x,x)=(x^2-1)^2$. Oct3 asked Will continuous extension preserve strict convexity? Sep21 accepted Is the limit point of this sequence unique? Sep21 comment Is the limit point of this sequence unique? Great. I'd better pick up a book and review some basics. Sep21 comment Is the limit point of this sequence unique? Thanks for your quick response. Please see my edit. Sep21 revised Is the limit point of this sequence unique? added 48 characters in body Sep21 asked Is the limit point of this sequence unique? Sep15 comment normal cone to sublevel set Without the assumption that ${\rm cone}\,\partial f(\bar{x})$ is closed, why is the inclusion $N_C(\bar{x})\subset{\rm cone}\,\partial f(\bar{x})$ equivalent to $(N_C(\bar{x}))^{\circ}\supset({\rm cone}\,\partial f(\bar{x}))^{\circ}$? Sep15 comment normal cone to sublevel set This is indeed an exercise in Borwein's book: Convex analysis and nonlinear optimization. Although it is an exercise, I think it is very useful to relate many stuff together. As I mentioned, it appears in many other textbooks as a theorem in a bit different form. Sep15 comment normal cone to sublevel set Moreover, I do not think a direction polar to all subgradients has to be a descent direction. Consider $f(x,y)=\frac{1}{2}(x^2+y^2)$ at $(1,0)$. The gradient is $(1,0)$. The direction $(0,1)$ is polar to $(1,0)$ and belong to the tangent cone. But it is indeed an ascent direction. Please let me know if I miss anything. Thanks. Sep15 comment normal cone to sublevel set Hi, gerw: could you please provide more detail about the last inequality? By the way, I think you also make use of the fact that ${\rm cone}\,\partial f(\bar{x})$ is closed. Thus, it is the same as $({\rm cone}\,\partial f(\bar{x}))^{\circ\circ}$. This is due to $0\notin\partial f(\bar{x})$. Am I right? Thanks a lot. Sep13 asked normal cone to sublevel set Sep1 accepted Is the smallest singular value able to measure the similarity between two matrices?