>function f: R^n --> R where for all x in R^n f(x) = 0
This hyperplane is not convex. A convex curve by definition can not be equal to its tangent at any point.
Edit: I should specify, I mean a convex curve cannot be completely equal to any of its tangents, obviously it will equal each tangent at a single point.
You don't want to consider just "tangents" and, instead, consider what I defined as supporting hyperplanes of the epigraph and subgradients of the function. If the gradient exists, that is, if the function is differentiable, then the subgradient really is a tangent. Otherwise can have many different subgradients supporting at one point on the curve and its epigraph.
It's simple: A cube has supporting planes at each point that is an edge or corner, but those points do not have tangents.
It sounds like you are describing curves that are strictly convex. Curves that are convex, but not strictly convex, can intersect their tangents at more than one point, or even at every point.
I'm going by the definition of convex function given in Rudin's "Principles of Mathematical Analysis", Apostol's "Calculus", Wikipedia, and MathWorld.
>function f: R^n --> R where for all x in R^n f(x) = 0
This hyperplane is not convex. A convex curve by definition can not be equal to its tangent at any point.
Edit: I should specify, I mean a convex curve cannot be completely equal to any of its tangents, obviously it will equal each tangent at a single point.