Let $C$ be the convex hull of the points $A = (1,0), B= (0,1)$ and $C=(-1,0)$.
Determine the set of points $P \in C$ for which there is a supporting hyperplane of $C$ in
$P$ and the points for which there is a supporting hyperplane that is unique.
For each point in $P$ for which there is a unique supporting hyperplane in $P$, give an equation of this hyperplane.
Clearly the convex hull is represented by a triangle, but I don't exactly know how to proceed to find the $P$ and the equations of the unique hyperplanes.
Are the points where there is a unique hyperplane not the points $A, B$ and $C$?
In $A$ for example, there is a unique supporting hyperplane with the equation: $x = 1$. I don't know if this is totally wrong…
Best Answer
The supporting hyperplanes in $A$, $B$ and $C$ are not unique, as multiple straight lines ($\Bbb R^2$ hyperplanes) can be drawn through those points, all of them suppoting $P$. For any point in the sides of the triangle, but not the vertices themselves, i.e., $$P'=\{\alpha A+\beta B+\gamma C\mid1>\alpha,\beta,\gamma\geq 0\,\wedge\,\alpha\beta\gamma=0\,\wedge\alpha+\beta+\gamma=1\},$$ there's a unique supporting hyperplane. The equations for them are just those of the lines that go through $A$ and $B$, $B$ and $C$ and $A$ and $C$.