# Continuity of Busemann-Hausdorff area density

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.

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).