# Continuity of a convex function on a vector bundle

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.

• 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. Mar 27, 2017 at 12:50
• 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. Mar 27, 2017 at 13:55
• 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" Mar 27, 2017 at 13:58
• 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 $n$. Mar 28, 2017 at 8:15
• 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. May 26, 2017 at 11:57