What is the name of $\hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\}$ for $K$ convex subset of a Hilbert space?

Question

Suppose I have a convex set $K\subset X$, where $X$ is say a real Hilbert space (for simplicity). Then, given some $a\in \Bbb{R}$, let

$$ \hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\} $$

This is seemingly very similar to the idea of a polar set or polar cone, but I can't quite find a definition of this exact object.

Answer 1

I have a very similar reference: The German "Funktionalanalysis" (functional analysis) by D. Werner: He defines for a (complex) locally convex topological vector space $X$ with dual $X'$: $$A^\circ = \{ x' \in X' : \Re \langle x' , x\rangle \le 1 \; \quad\forall x \in A\}.$$ (In the real case, you can simply drop the "$\Re$".) He calls this set "Polare", I would translate this as "polar set". Note that if $A$ is a cone, this is indeed the polar cone.

Note that you can pull out the $a$ in your definition. Indeed, we have $$\hat K = a \, K^\circ.$$