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

necessarycondition (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

necessarycondition (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 $B$.

closedconvex set in R^n is always envelope of tangent half-spaces, which does not serve as a necessary condition to my OP since I did not put closedness in OP. Appreciated. $\endgroup$3more comments