*Is it true that the length spectrum of a compact Riemannian manifold determines its volume?*

This question was inspired by the MO question Length spectrum of spheres . BS's answer recalls a theorem of Duistermaat and Guillemin (Duistermaat, J. J.; Guillemin, V. W. (1975), "The spectrum of positive elliptic operators and periodic bicharacteristics", Inventiones Mathematicae 29 (1): 39–79) that states that for generic metrics the length spectrum and the spectrum of the Laplacian determine each other. Since Weyl's asymptotic formula implies that the spectrum of the Laplacian determines the volume, it may *seem* that the length spectrum determines the volume. Is this right?

**Addendum.**
This question seems more useful interesting and useful if it is reformulated as follows: *what does the set of lengths of periodic geodesics and/or its various analogues such as the length spectrum and the marked length spectrum say about the volume of a compact Riemannian or Finsler manifold?*

Here are more concrete questions around this topic:

**1.** Let $M$ be a simply-connected, compact $n$-manifold. Does there exist a quantity $c(M) > 0$ such that for any Riemannian metric $g$ on $M$ the volume of $(M,g)$ is bounded below by $c(M)$ times the $n$-th power of the length of the shortest periodic geodesic of $(M,g)$?

This is a really hard open problem. It was solved by Croke for the $2$-sphere in
*Area and the length of the shortest closed geodesic* J. Differential Geom. Volume 27, Number 1 (1988), 1-21. The current record for $c(S^2)$ is held by Rotman:
*The length of a shortest closed geodesic and the area of a -dimensional sphere* Proc. Amer. Math. Soc. 134 (2006), 3041-3047. However, the conjectured optimal constant $1/2\sqrt{3}$
is still a challenge.

**2.** Are there known (explicit?) examples of compact Riemannian manifolds with the same set of lengths of periodic geodesics (the *length set*) and different volume?

BS gave a reference (see his answer) where Huber shows that two compact hyperbolic surfaces with the same length spectrum have the same laplace spectrum and hence the same volume. For negatively curved, compact surfaces J.-P. Otal showed that the **marked** length spectrum determines the surface up to isometries. This is not true for Finsler metrics (they are much more flexible than Riemannian metrics), however:

**3.** If two negatively-curved, compact Finsler surfaces have the same marked spectrum, do they also have the same area?

However, posing these type of question is easy. I'm more interested in collecting some success stories.