I'm reading this paper [1]. To get some intuition on what a geodesically convex function is like, I'm trying to find some example in $GL_n(\mathbb{R})^+$ that is geodesically convex (near $I$).
Let me first copy some definitions from [1].
Definition 2 [1]. A function $f : \mathcal{M} \rightarrow \mathbb{R}$ is said to be geodesically convex if for any $x, y \in \mathcal{M}$, a geodesic $\gamma$ such that $\gamma(0) = x$ and $\gamma(1) = y$, and $t \in [0, 1]$, it holds that $$f(\gamma(t)) \le (1 − t) f(x) + t f(y).$$ It can be shown that an equivalent definition is that for any $x, y \in \mathcal{M}$, $$ f(y) \ge f(x) + \left< g_x, Exp_x^{-1} (y) \right>_{x}, $$ where $g_x$ is a subgradient of $f$ at $x$, or the gradient if $f$ is differentiable, and $\left< \cdot , \cdot \right>_x $ denotes the inner product in the tangent space at $x$ induced by the Riemannian metric.
Definition 3 [1]. A function $f : \mathcal{M} \rightarrow \mathbb{R}$ is said to be geodesically $\mu$-strongly convex if for any $x, y \in \mathcal{M}$, $$ f (y) \ge f (x) + \left< g_x, Exp_x^{-1} (y) \right>_{x} + \frac{\mu}{2} d^2 (x,y), $$ [where $d$ is the distance between $x$ and $y$ induced by the Riemannian metric.]
Based on the definition of geodesic convexity and strong geodesic convexity, I'm guessing, in $GL_n (\mathbb{R})^+$, the function $f (x) = d^2(x,I)$ ($d (\cdot, \cdot)$ is the distance induced by some Riemann metric) is geodesically convexity near $I$.
I'm trying to use the (relatively common) Riemann metric $\left< \cdot, \cdot \right>_x$ on $GL_n (\mathbb{R})$ such that $ \left< A, B \right>_x = \left< x^{-1} A, x^{-1} B \right>_I $ and $ \left< A, B \right>_I = tr (A^\top B) $.
It seems, in the above mentioned setting, verifying $ f (x) = d^2 (x,I)$ fits into the convexity (or strong convexity) definitions may not be an easy task.
Are there some (clean) examples that fits into the definitions above on $GL_n (\mathbb{R})^+$ (near $I$)?
[1] H. Zhang and S. Sra, First-order Methods for Geodesically Convex Optimization, JMLR, 2016.
