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The recent results of An, Luan, and Yen [Differential stability in convex optimization via generalized polyhedrality. Vietnam J. Math. https://-doi.org/10.1007/s10013-024-00721-y] on differential stability of parametric optimization…
It is a survey of the results on the classification of stable homotopy types of polyhedra of small dimensions, mainly obtained by H.-J. Baues and the author. The proofs are based on the technique of matrix problems (bimodule categories).
We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…
Many engineering failures in orientation-dependent systems are geometric failure modes: changing the geometry can eliminate what changing the material merely delays. The mono-monostatic property (exactly one stable equilibrium under…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
It is a classical result of Euler that the rotation of a torque-free three-dimensional rigid body about the short or the long axis is stable, whereas the rotation about the middle axis is unstable. This result is generalized to the case of…
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…
We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature…
The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the…
In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only…
It is known that the $L_p$-curvature of a smooth, strictly convex body in $\mathbb{R}^{n}$ is constant only for origin-centred balls when $1\neq p>-n$, and only for balls when $p=1$. If $p=-n$, then the $L_{-n}$-curvature is constant only…
This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical,…
The purpose of the present paper is to show few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is…
The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form…
A convex polygon is defined as a sequence (V_0,...,V_{n-1}) of points on a plane such that the union of the edges [V_0,V_1],..., [V_{n-2},V_{n-1}], [V_{n-1},V_0] coincides with the boundary of the convex hull of the set of vertices…
In this Letter we suggest a method of convex rigid frames in the studies of the multipartite quNit pure-states. We illustrate what are the convex rigid frames and what is the method of convex rigid frames. As the applications we use this…
We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can…
It is known that each symmetric stable distribution in $R^d$ is related to a norm on $R^d$ that makes $R^d$ embeddable in $L_p([0,1])$. In case of a multivariate Cauchy distribution the unit ball in this norm corresponds is the polar set to…
This text is a somewhat reformatted (e.g., some statements that were not as such in the original paper, are given the names "Corollary" or "Theorem.") translation of the old and practically inaccessible paper: P. Kuchment, On the question…
We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal shape and the impedance parameter can be variable. We establish the stability results by…