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.
PS: You have found one of the finest math books ever written: Rockafellar-Wets's Variational Analysis - Have fun with it!