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Let $X$ be a Hilbert space. Normal cone is defined like this: For a convex set $C$, the associated normal cone $$N_C(x) = \{z \in X: \langle z, y-x\rangle \leq 0 \quad \ \forall \ \ y \in C\}$$ of $C$ at point $x \in C$.

Can someone tell me what is its physical interpretation. Why it is normal?

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    $\begingroup$ Try taking $X = \mathbb{R}^2$ or $\mathbb{R}^3$ and drawing a picture. $\endgroup$
    – Nate Eldredge
    May 2, 2017 at 16:53
  • $\begingroup$ Note that 'normal' is to be understood in the sense of 'orthogonal' or 'perpendicular'. $\endgroup$
    – gerw
    May 2, 2017 at 19:47

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This is often called the dual cone. Wikipedia has this illustration (here $x=0$ and $C$ is neither convex nor contains $x$, but the concept is the same):

dual cone

Think of the dual cone $C^*$ as the set of directions such that starting from $x$, you don't move away from any point of $C$ (initially). When $C$ is itself a cone, $C^*$ consists of the linear functionals that are bounded below on $C$. This suggests that the concept may be relevant in linear programming and it is, as the article Conic optimization indicates.

Dual/normal cone is the same thing in the world of convex cones as polar set is in the world of convex sets.

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