Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I want to discuss the uniqueness of the supporting hyperplane.
A special situation where uniqueness holds is when the supporting hyperplane at $p\in\partial C$ coincides with the tangent pf $\partial C$ at $p$. For example, when the $\partial C$ is piecewise linear, then the supporting hyperplanes must coincide with tangent planes except at vertices of $C$.
There are sufficient conditions, say $C$ being strictly convex, which guarantees the uniqueness of the supporting hyperplane. But strict convexity does not lead to the conclusion that the supporting hyperplane coincide with tangents. Moreover. since the supporting hyperplane theorem is no more than Hahn-Banach theorem, I was wondering
(i)Is there a necessary condition (on $\partial C$, say some sort of
algebraic regularity; conditions on $C$ are also fine but less interesting) about the
boundary of convex set $C$ to make the supporting hyperplane unique?(ii)Is there a necessary condition (on $\partial C$, say some sort of algebraic regularity) about the
boundary of convex set $C$ such that the supporting hyperplane of $C$
at any point of $p\in\partial C$ coincide with the tangent plane of
$\partial C$ at $p$?
For my purpose, I only want to know the case $C\subset\mathbb{R}^n$ with the usual topology.
($C$ does not have to be closed convex set but only convex, closedness is too strong for my purpose.)
Additionally, how will the situation changes if $C\subset H$ for a general (infinite dimensional) Hilbert space? Or even more general, in a Banach space $B$.
Best Answer
I think it goes like this: Assume $C$ has nonempty interior. Rotating the convex set if needed, the boundary $\partial C$ near any given point $p \in \partial C$ can be written as the graph of a convex function. More or less by definition, a hyperplane is a supporting hyperplane at $p$ if and only if it corresponds to an element of the subdifferential of $f$. Moreover, $f$ is differentiable at $p$ if and only if the subdifferential contains only one element. In that case, the unique supporting hyperplane is the tangent hyperplane of $\partial C$ at $p$.
In summary, a supporting hyperplane is the tangent hyperplane at $p \in\partial C$ if and only if $f$ is differentiable at $p$ if and only if the subdifferential of $f$ at $p$ contains only one element if and only if there is only one supporting hyperplane at $p$.
This, I believe, follows by results stated and proved in the book Convex Analysis by Rockafellar.
EDIT: As HenryL points out, this holds only if the supporting hyperplane actually touches the boundary. If $C$ is noncompact, then it is possible this does not happen.