TITLE: Finsler length is lower semicontinuous
QUESTION [6 upvotes]: Edit: Corrected the title. 
For a smooth (or say, Lipschiz) curve $\gamma: [a, b] \to \mathbb{R}^n$, define
$$L(\gamma) := \int_a^b f(\gamma’(t), \gamma(t)) dt$$
where $f(\cdot, x)$ is a norm (in the variable $\cdot$) and $f$ is continuous. Prove that L is lower semicontinuous with respect to pointwise convergence, meaning if $\gamma_i$ is a sequence of curves converging to $\gamma$, then
$$\liminf_{i \to \infty} L(\gamma_i) \ge L(\gamma).$$
I could prove this for the ordinary Euclidean length:
$$L(\gamma) = \int_a^b |\gamma’(t)| dt$$
but I can’t seem to deal with the case that the function $f$ depends on the second variable so a similar argument to when this is the ordinary Euclidean length of a curve isn’t working here. In essence, I’ve been having trouble dealing with the fact that something like the following might not hold: $f(v + w, x) \le f(v, x) + f(w, x')$.
Edit: Perhaps I should mention: I got this from exercise 2.4.5 (and exercise 5.1.1) of A Course in Metric Geometry by Burago, Burago, and Ivanov. The part that I'm getting is when $f$ does not depend at all on its second variable, but a similar approach I'm using is not working for when $f$ depends on its second variable.
Edit: Fixed some typos.
Edit: changed “seminorm” to “norm” since, as pointed out below, the statement is incorrect for “seminorm.”

REPLY [3 votes]: It's not enough to assume $f(\cdot,x)$ is a seminorm - note that a Finsler length structure requires a norm. Consider
\begin{align*}
f(p,x)&=|p_1x_2-p_2x_1+p_3|\\
\gamma(t)&=(0,0,t)&&(0\leq t\leq 1)\\
f(\gamma'(t),\gamma(t))&=1\\
\gamma_i(t)&=(i^{-1}\cos(ti^2),i^{-1}\sin(ti^2),t)\\
\gamma'_i(t)&=(-i\sin(ti^2),i\cos(ti^2),1)\\
f(\gamma'_i(t),\gamma_i(t))&=0
\end{align*}
So $\gamma_i$ converges uniformly to $\gamma,$ but $L(\gamma_i)=0<1=L(\gamma).$
If $f(\cdot,x)$ is a norm, given uniform convergence of $\gamma_i$ you can use an estimate like $(\forall\epsilon>0)(\exists\delta>0)\|x-y\|\leq\delta\implies f(p,x)\geq (1-\epsilon)f(p,y)$ to locally approximate $f$ by a constant norm. So the argument should be roughly like the Euclidean case.