Continuity of Busemann-Hausdorff area density

Question

I am trying to find out why the Busemann-Hausdorff area density as defined by Burago and Ivanov is continuous. Here, $GC_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional vector space $V$, that is, vectors $\sigma = v_1 \wedge \cdots \wedge v_m$. The BH-density is given by $$ A^{bh}: GC_m(V) \to \mathbb{R}, \sigma \mapsto \frac{\varepsilon_m}{\mathcal{H}^m(L_\sigma^{-1}(B))} $$ where $\sigma=v_1\wedge\cdots\wedge v_m$ and $L_\sigma:\mathbb{R}^m \to V, e_i \mapsto v_i$ maps the standard basis linearly to the span of the $v_i$. I can show that this is both well-defined (because $L_\sigma$ is not well-defined as function of $\sigma$) and absolutely homogeneous but can't figure out the continuity.

First, I am wondering which topology to understand the continuity in. I assumed that it's the norm topology of $\Lambda^m(V)$ but that's a bit strange because I need to assume that $V$ has in inner product (but the problem comes from Finsler geometry).

Secondly, all references I found - such as Álvarez/Thompson's "Volumes on Normed and Finsler spaces" or Burago and Ivanov's "Minimality of planes in normed spaces" (link above) - do not prove the continuity of this density nor answer my first question. I think I might overlook something very simple.

I asked this on math.SE already but I thought it might be too specific. Tell me and I'll delete this question here if it is too simple for MO.

Answer 1

Yes, it is continuous (and if the Finsler metric is smooth, it is also smooth). All you have to do to see it is to fix some Riemannian metric (or local coordinates) and verify that the volume of the unit tangent balls varies continuously (resp. smoothly) with the base point. The same goes for the Holmes-Thompson volume density (work with the dual balls in this case). Continuity is true for all notions of volume as defined in the Alvarez-Thompson paper (this follows from the monotonicity axiom). Smoothness is not true for every definition: for example Gromov's mass* volume density of a smooth Finsler metric is not necessarily smooth.

Back to Busemann-Hausdorff, it's just a basic fact for parametric families of convex bodies: if you have a family of norms on $\mathbb{R^n}$ depending continuously on some parameters, the volume of the unit balls, as a function of the parameters, will be continuous (just use the integral formula for the volume of a star-shaped set that uses the "norm" or gauge and see that you get an integral with parameters).