Consider the following energy function $$-\sum_{i<j\in[n]}\cos(\theta_i-\theta_j)$$ where $\theta_i\in\mathbb{R}$, for $\forall i\in[n]$.
At any vector $\vec\theta$ that is not all-equal vector ($\theta_i=\theta_j,\forall i<j\in[n]$), move it to the point $\theta^*$ such that for any $i,j\in[n]$, $$\theta^*_i-\theta^*_j=t(\theta_i-\theta_j)$$ where $t\in(0,1]$. It is clear that such $\theta^*$ are not unique. For example, take $n=3$ and set $\vec\theta=(0,\pi/6,\pi/3)$. Then $\vec\theta^*$ can be $(0,t\pi/6 ,t\pi/3)$ or $(a,t\pi/6 +a,t\pi/3 +a)$ for any real number $a$.
Thus, the direction from point $\theta$ to $\theta^*$ is not unique.
My question is, does the steepest descent direction is among those directions $\theta\rightarrow\theta^*$?
My intuition is all those directions are not the steepest direction on the energy function. This is because these directions are the straight line connecting $\theta$ and all-one vectors. And the straight line are perpendicular to the level set of another function $\max_{i<j\in[n]}\{|\theta_i-\theta_j|\}$, if and only if the direction $\theta\rightarrow\theta^*$ is the steepest direction (ascent or descent) of $\max_{i<j\in[n]}\{|\theta_i-\theta_j|\}$. However, clearly $$\max_{i<j\in[n]}\{|\theta_i-\theta_j|\}\neq -\sum\cos(\theta_i-\theta_j)$$ thus those direction cannot be the steepest direction of $-\sum\cos(\theta_i-\theta_j)$.