$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a concave function with local maximum at $\mathbf{x}^*$ in a convex, closed feasible set $\mathcal{F}\subset\mathbb{R}^n$. Now consider a suboptimal point $\mathbf{x}\in\mathcal{F}$. If we move from $\mathbf{x}$ in such a direction (within the feasible region) that
1) $\mathcal{L}_1$ norm of $\mathbf{x}-\mathbf{x}^*$ increases monotonically.
2) $\mathcal{L}_2$ norm of $\mathbf{x}-\mathbf{x}^*$ increases monotonically.
Does it mean $f\left(\mathbf{x}\right)$ will decrease in both of the above cases (considered independently)?
Edit: No. Consider $f(x,y) = -(x^2+9y^2)$ defined on $\mathbb{R}^2$. The origin is the sole local and global maximum point of $f$. If we move from $(1,1)$ to $(3,0)$ along the line $x+2y-3=0$, the $L_1$ norm increases from $2$ to $3$ and the $L_2$ norms increases from $\sqrt{2}$ to $2$, yet $f(1,1)=-10<f(3,0)=-9$. If you plot a graph, you will see that $f$ increases monotonically at first, then attains the maximum value $-6.23$ at $(x,y)\approx(2.08,0.46)$ in the course of movement and then decreases monotonically to $-9$.