$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(3)$, represented with $3 \times 3$ rotation matrices, equipped with the bi-invariant metric $$\langle \xi, \eta \rangle = \frac{1}{2} tr(\xi^T \eta).$$ Here, $\xi,\eta \in \mathfrak{so}(3)$ are skew-symmetric. During some research, I came across the following inequality I wanted to prove:

$$\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle$$ for all $R,S \in \SO(3)$ where the above inequality is well-defined. A similar, yet stronger, inequality I noticed was $$\langle \log(R), \log(R^{-1}S)\rangle + \frac{1}{2} \|\log(R^{-1}S)\|^2 \geq \langle \log(R), \log(S)-\log(R) \rangle + \frac{1}{2} \|\log(S) - \log(R)\|^2.$$

I found this quite unintuitive due to the fact that $\|\log(R^{-1}S)\| \leq \|\log(S) - \log(R)\|$. As a result, I would expect the inequality to be flipped. But every pair of rotation matrices I generated, this inequality held.

My attempt at a proof was utilizing the fact that geodesic distance is $\mu$-strongly geodesically convex, and hence is lower bounded by its 2nd order approximation (this was how I originally came across these inequalities).

By strongly convex, I mean the following. Let $(M,g)$ be a complete Riemannian manifold. A differentiable function $f:Q \subset M \to \mathbb{R}$ is called $\mu$*-strongly convex* on $(M,g)$ if $Q$ is geodesically convex and for all $x,y \in Q$, we have $$f(x) \geq f(y) + \langle \nabla f(y), \log_y(x)\rangle_y + \frac{\mu}{2}\|\log_y(x)\|^2_y.$$

Here, a subset $Q \subset M$ is called *geodesically convex* if for all $x,y \in Q$, there exists a **unique** **minimizing** geodesic connecting $x$ to $y$, and that geodesic is contained in $Q$.

If we fix $y \in M$ and set $f(x):=\frac{1}{2}d^2(x,y)$, one can show that $f$ is $\mu$-strongly geodesically convex on $B \subset M$, where $B$ is a geodesically convex geodesic ball centered at $y$. Here, $\mu$ also has a closed form solution if $M$ has bounded above sectional curvature.

However, I was not able to prove any of these. Any ideas?

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