# Some tricky details about PCA non-convexity

I am reading about PCA and I came up with a some contradictions. PCA based on this post, PCA optimization problem is given by:

\begin{aligned} \max_{w} \quad & w^T\Sigma w\\ \textrm{s.t.} \quad & w^Tw=1\\ \end{aligned}\quad(\text{P1}) where $$\Sigma$$ is the covariance matrix of data. My questions are:

1. In this post, it was stated by @whuber that $$w^T\Sigma w$$ is not Convex. How is that possible as $$\nabla w^T\Sigma w=\Sigma \succeq 0$$ (link)?
2. Also, in the same post it is stated that the unit sphere $$S^{n-1}\subset\mathbb{R}^n$$ is decidedly not convex. How it is that possible as it is proven that $$w^Tw=1$$ is convex?
3. Also, if $$w^T\Sigma w$$ is convex what is the intuition of maximizing it?

Could you please some one shade some light?

EDIT: I removed the second question. I just realized that we are dealing with a non-convex domain which cause as a non-convex problem, that is why we maximize (P1). Please, correct me If I am wrong.

1. $$w^T\Sigma w$$ is convex function, you're right. As far as I see, whuber's answer defines concave functions as what we usually know as convex functions. It's also pointed out in the comments.