What are good examples of polar sets in $\mathbb R^2$? Let $C\subset\mathbb{R}^n$ non-empty.
Its polar set $C^\circ$ is defiend as
$$C^\circ\equiv \{y\in \mathbb{R}^n\mid \forall x\in C, \,\,\langle y,x\rangle  \le 1 \},$$
where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product.
I cannot think of an example of a set and its polar set in $\mathbb{R}^2$. What is a good example of a set and its polar set?
 A: A few examples include:

*

*The polar set of a singleton $\{x\}$ is the halfspace. If $\|x\|$ then the halfspace touches $x$, otherwise its below $x$ iff $\|x\|>1$:

$\hspace{40pt}$


*The polar set of a unit disk is the disk itself.


*See this question for the polar set of $\{(x,y)\in\mathbb R^2: x^2+y^4\le1\}$.


*Consider the polytope with the four vertices $(\pm 1,\pm1)$ (a square centred at the origin). Then its polar set is a rotated square with vertices $(0,\pm1),(\pm1,0)$:
$\hspace{40pt}$


*You can find more examples in these lecture notes by Jonathan Kelner (Link to pdf).
If you have access to Mathematica, I made a simple snippet to explore how the polar sets of finite numbers of points look like. Just click anywhere to add points and see the corresponding polar set:
$\hspace{40pt}$
Here's the code to generate this visualisation:
DynamicModule[{pts={{1,1}}},
    EventHandler[
        Dynamic@Show[
            Graphics[{
                PointSize@0.04,Dynamic@Point@pts,
                Circle[]
            },
                Axes->True,AxesOrigin->{0,0},Frame->False,PlotRange->ConstantArray[{-2,2},2],
                GridLines->Automatic,AxesStyle->Directive[Large,Black]
            ],
            RegionPlot[
                And@@(Dot[#,{x,y}]<=1&/@pts),
                {x,-4,4},{y,-4,4},PlotPoints->50
            ]
        ],
        {"MouseClicked":>(AppendTo[pts,MousePosition["Graphics"]])}
    ]
]


See source of this answer for the MMA code used to generate the figures.
A: The basic example given by LinAlg in the comments is the key to understanding polar sets in general: when $C=\{c\}$ is just a point, then
$$
C^\circ = \{\, y\in\mathbb R^n \,|\, \langle y, c\rangle \le 1\,\} =: H_c
$$
is a half-space with the boundary hyperplanes outward normal direction being the direction of $c$ and distance from the origin given by $1/\|c\|$.
Using this, we can express polar sets in the following way for arbitrary $C\subseteq \mathbb R^n$:
$$
C^\circ = \{ \,y\in \mathbb R^n \,|\, \forall c\in C\colon y\in H_c\,\} = \bigcap_{c\in C} H_c.
$$
Hence, the polar set is an intersection of all the half-spaces $H_c$ where $c$ ranges over the points in $C$.
