In this survey article we will consider universal lower bounds on the volume of a Riemannian
manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of …

The systole of a compact metric space $ X $ is a metric invariant of $ X $, defined as the
least length of a noncontractible loop in $ X $. When $ X $ is a graph, the invariant is usually …

We say that a Riemannian manifold (M, g) with a non-empty boundary∂ M is a minimal
orientable filling if, for every compact orientable (M̃, g̃) with∂ M̃=∂ M, the inequality d g̃ (x …

The study of volumes and areas on normed and Finsler spaces is a relatively new field that
comprises and unifies large domains of convexity, geometric tomography, and integral …

We construct a class of Finsler metrics in three-dimensional space such that all their
geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This …

We explore a natural generalization of systolic geometry to Finsler metrics and optical
hypersurfaces with special emphasis on its relation to the Mahler conjecture and the …

Our purpose here is to give an overview of known results and open questions concerning
the volume product 𝒫 (K)= minz∈ K vol (K) vol ((K− z)∗) of a convex body K in ℝn. We …

The main" unconditional" result of this paper, Theorem 3, states that every two-dimensional
affine disc in a normed space (that is, a disc contai in a two-dimensional affine subspace) is …

Recently, Balacheff ['A local optimal diastolic inequality on the two‐sphere', J. Topol. Anal. 2
(2010) 109–121] proved that the Calabi–Croke sphere made of two flat 1‐unit‐side …

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal
is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This …