Consider the rank-${n \choose m}$ vector bundle $\pi\colon E:=\bigwedge^m(TN)\to N$ over a smooth Finsler manifold $N$ and equip each fibre $E_q := \pi^{-1}(q)$ with a norm that depends smoothly on $q\in N$ (induced by an arbitrary Riemannian metric on $N$).

Let $A\colon \bigwedge^m(TN)\to \mathbb{R}$ be a function with the property that the restriction to each fibre $E_q$ is a convex and absolutely homogeneous extension of a continuous function $a_q:GC_m(T_qN)\to \mathbb{R}$. Here $GC_m(V)$ is the set of simple or totally decomposable $m$-vectors in $V$. More precisely, I am looking at the $m$-dimensional Busemann--Hausdorff area density as defined in [Section 4.1, p. 18] and its extension (for $m=2$ in [Theorem 1, p. 3]).

How can I show that $A$ is continuous on $E$?

Indeed, I have shown that $A_q := A\big|_{E_q}$ is continuous on each fibre by using that bounded, convex functions are locally Lipschitz (e.g. [Clarke-Optimization and nonsmooth analysis, Proposition 2.2.6, p. 34]). I would like to compare elements from different fibres now to show continuity on the complete total space $E$. Intuitively it feels like it is clear because the fibres vary smoothly with $q$ (because $N$ is smooth) and on each fibre $A$ is continuous. I fail to make this intuition rigorous using local trivialisations though.

Any hint or comment is welcome.

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    $\begingroup$ Is that true? Take a Finsler metric seen as a function on the tangent bundle and multiply it by the pullback of discontinuous function at the base. The resulting function will not be continuous on the tangent bundle. $\endgroup$ Mar 27, 2017 at 12:50
  • $\begingroup$ You are right. I tried to write my questions in a fairly general form but must have missed something - I will update my question. Actually the function $A$ in my case is a convex extension of the $m$-dimensional Busemann--Hausdorff area density of a Finsler manifold (which exists for m=2 due to Burago/Ivanov). So the vector bundle in my case is $\bigwedge^m(TN)$ with the norm on each fibre coming from the inner product induced by any Riemannian metric on the Finsler manifold $N$. We talked about this last summer in Leipzig but I still fail to write it down unfortunately. $\endgroup$ Mar 27, 2017 at 13:55
  • $\begingroup$ In fact, for the BH-density defined on the simple $m$-vectors I still do not really understand what the topology of $GC_m(TN)$ is, so that one can speak of continuous functions as in your survey article "Volumes on Normed and Finsler spaces, Def. 4.2, p.18" $\endgroup$ Mar 27, 2017 at 13:58
  • $\begingroup$ The fancy description of the topology on $GC_m(TN)$ is that this is just a line bundle (the determinant line bundle) over the Grassmannian bundle of $m$-dimensional tangent planes. The pedestrian definition is that a function $f$ defined on this space is continuous at a point $(n,v_1 \wedge \cdots \wedge v_m)$ if and only if for any $m$ continuous and linearly-independent vector fields $X_1, \ldots, X_m$ defined on a nbd of $n \in N$ and for which $X_1(n) \wedge \cdots \wedge X_m(n) = v_1 \wedge \cdots \wedge v_m$, the function $f(X_1(x) \wedge \cdots X_m(x))$ is continuous at $n$. $\endgroup$ Mar 28, 2017 at 8:15
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    $\begingroup$ No, of course I don't mind. I do need this form of continuity because I want to use the lower-semicontinuity result of Acerbi-Fusco (wpage.unina.it/nfusco/paper6.pdf#page=8) for the functional $area(X) = \int_B A(X(p),X_{p_1} \wedge X_{p_2}) dp$. These kind of lsc-results need the continuity of the integrand on the product $R^n \times R^N$ or in my case on the whole vector bundle. Weaker assumptions lead to counter-examples according to Acerbi-Fusco. $\endgroup$ May 26, 2017 at 11:57


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