# Is the normal cone of a polyhedron a set of points or a set of vectors?

I looked for the normal cone to a polyhedron, and I found this definition from https://sites.math.washington.edu/~rtr/papers/rtr169-VarAnalysis-RockWets.pdf:

But this is confusing to me. it seems to me like the $\sum y_i a_i$ will yield a single value and not a vector. it was my understanding that the normal cone is a set of vectors.

Is my understanding wrong? Or am I misinterpreting what this book is saying for the Normal cone of a polyhedron?? thank.

Note that on the right side of your formula for $N_C(\bar{x})$, there are curly brackets indicating this is a set. It is a special set of linear combinations with all coefficients being nonnegative. It is a finitely generated convex cone.
• I see, the $a_i$'s are actually vectors, NOT elements of a single vector. thank you @max_zorn. – jaja May 6 '18 at 6:55
• Although, technically $a_i$ is a row vector. So, I imagine they should actually all be transposed in order for the result of $\sum y_i a_i$ to be a column vector. i.e. I think it should be $\sum y_i a_i^T$. do you think so too @max_zorn? – jaja May 6 '18 at 18:19