Post Closed as "not a real question" by Benjamin Steinberg, Igor Rivin, Martin Brandenburg, alvarezpaiva, Ryan Budney
fundamental group is non-trivial, but in some sense it is small (i.e., the manifold is not essential in the sense of Gromov). However, there may always be short contractible geodesics. volume of all cones formed in this way is always the same, then the body is an ellipsoid.

Additional problem: I had forgotten another beautiful problem from the paper of Busemann and Petty: Problems on convex bodies, Mathematica Scandinavica 4: 88–94.

Minimality of flats in normed spaces. Given a closed $k$-dimensional polyhedron in an $n$-dimensional normed space with $n > k$, is it true that the area (taken as $k$-dimensional Hausdorff measure) of any facet does not exceed the sum of the areas of the remaining facets?

• When $n = k + 1$ this is a celebrated theorem of Busemann, which convex geometers are more likely to recognize in the following form: the intersection body of a centrally symmetric convex body is convex. A nice proof and a deep extension of this theorem was given by G. Berck in Convexity of Lp-intersection bodies, Adv. Math. 222 (2009), 920-936.
• When $k = 2$ this has "just" been proved by D. Burago and S. Ivanov: http://front.math.ucdavis.edu/1204.1543
• It is not true that totally geodesic submanifolds of a Finsler space (or a length metric space) are minimal for the Hausdorff measure. Berck and I gave a counter-example in What is wrong with the Hausdorff measure in Finsler spaces, Advances in Mathematics, vol. 204, no. 2, pp. 647-663, 2006.
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Post Made Community Wiki by alvarezpaiva
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