Related papers: The Two Hyperplane Conjecture
We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved…
This report addresses the solution of Riemann problems for hyperbolic equations when the nonlinear characteristic fields loose their genuine nonlinearity. In this context, exact solvers for nonconvex 1D Riemann problems are developed. First…
We propose to study the restriction conjecture using decoupling theorems and two-ends Furstenberg inequalities. Specifically, we pose a two-ends Furstenberg conjecture, which implies the restriction conjecture. As evidence, we prove this…
The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of…
A small variation of the circular shape of the hodograph theorem states that for every elliptical solution of the two-body problem, it is possible to find an appropriate inertial frame such that the speed of the bodies is constant. We use…
We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for homogeneous ideals in polynomial rings. Our theorem allows us to give…
Often some interesting or simply curious points are left out when developing a theory. It seems that one of them is the existence of an upper bound for the fraction of area of a convex and closed plane area lying outside a circle with which…
The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in $\mathbb R^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it…
An inverse problem of elasticity of $n$ elastic inclusions embedded into an elastic half-plane is analyzed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the…
To clarify some aspects of the application of Special Relativity, spacetime is sliced into null geodesic hypersurfaces as an alternative to the hypersurfaces of simultaneity normally adopted. Events at particle locations on the hypersurface…
We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value…
This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler…
In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary…
We determine the maximal hyperplane sections of the regular $n$-simplex, if the distance of the hyperplane to the centroid is fairly large, i.e. larger than the distance of the centroid to the midpoint of edges. Similar results for the…
We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if every…
We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the…
We consider the motion of a particle along the geodesic lines of the Poincar\`e half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version…
A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic…
In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross…
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…