Related papers: Inradius estimates for convex domains in 2-dimensi…

In this note, we study the radius of positively curved or non-negatively curved Alexandrov space with strictly convex boundary, with convexity measured by the Base-Angle defined by Alexander and Bishop. We also estimate the volume of the…

We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.

Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is…

We obtain sharp volume bounds on the boundaries of Alexandrov spaces with given lower curvature bound, dimension, and radius. We also completely classify the rigidity case and analyze almost rigidity. Our results are new even for smooth…

In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the…

We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to…

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci…

We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere, provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for…

Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \frac{\pi}{2}. We exhibit many such spaces that…

A Gauss equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature less than or equal to A, defined by a cubic inequality on the difference of arc and chord, has…

A sharp lower bound for the injectivity radius in noncompact nonnegatively curved Riemannian manifolds involving their soul goes back to \v{S}arafutdinov. We generalize this bound to the setting of Alexandrov spaces. Our main theorem reads…

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute…

We give a sharp lower bound on the area of a domain that can be enclosed by a closed embedded $\lambda$-convex curve of a given length on the Lobachevsky plane.

In this paper we give a new proof for an almost isometry theorem in Alexandrov spaces with curvature bounded below.

We show that if an Alexandrov space $X$ has an Alexandrov subspace $\bar \Omega$ of the same dimension disjoint from the boundary of $X$, then the topological boundary of $\bar \Omega$ coincides with its Alexandrov boundary. Similarly, if a…

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero…

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed…

We explore the Steklov eigenvalue problem on convex polygons, focusing mainly on the inverse Steklov problem. Our primary finding reveals that, for almost all convex polygonal domains, there exist at most finitely many non-congruent domains…

Let $M$ be an $n$-dimensional Alexandrov space with curvature $\geq 1$, and let $\{q_1,\cdots,q_k\}$ be any $\frac\pi2$-separated subset in $M$ (i.e. the distance $|q_iq_j|\geq\frac{\pi}{2}$ for any $i\neq j$). Under the additional…

Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in…