# Gradient always points away from the minima of convex functions?

I'm reading a paper by Léon Bottou, "Online Learning and Stochastic Approximations". He studies online learning with cost functions $$C(w)$$ satisfying two conditions:

1. $$C(w)$$ has a single minimum $$w^*$$.
2. $$C(w)$$ satisfies $$\inf_{||w-w^*||>\epsilon} (w-w^*)\cdot\nabla_w C(w) > 0$$ for all $$\epsilon>0$$ and all $$w$$ in its domain. In words, the gradient always points away from the minimum.

Here $$w$$ is a vector in $$\mathbb R^n$$.

The author states that "this condition is weaker than the usual notion of convexity." But it's not completely clear. My question is this:

Do all strictly convex functions $$C(w)$$ satisfy these conditions?

The first condition is trivial since all strictly convex functions have a single minimum (excluding cases where the extrema are at the boundary of the domain). So my question is really about the second condition.

• "...since all convex functions have a single minimum". The convex function $f(x,y)=x^2$ has the whole line of minima ($x=0$). – A.Γ. Jun 18 at 13:32
• @A.Γ. Sorry. I meant strictly convex functions. I edited the question. – becko Jun 18 at 13:49

Not all strictly convex functions have minimum, but assume it does. Then it is unique, true. Furthermore, the expression $$(w^*-w)\cdot\nabla_w C(w)$$ is a scalar product between the vector from the point $$w$$ to the minimum and the gradient at the point. The gradient is orthogonal to the level set $$\{x\colon f(x)=f(w)\}$$ and points outwards, the sublevel set $$\{x\colon f(x)\le f(w)\}$$ is convex, and the minimum $$w^*$$ is in the interior of the sublevel set as $$f(w^*). It means that the gradient and the vector $$w^*-w$$ point strictly at the opposite half-spaces w.r.t. the tangent hyperplane. Hence, yes, if a strictly convex function is differentiable then it does satisfy $$-(w^*-w)\cdot\nabla_w C(w)>0$$.