Related papers: Problems of maximal mean resistance on the plane

We show that in all dimensions d>2, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.

The method of Hessian measures is used to find the differential equation that defines the optimal shape of nonrotationally symmetric bodies with minimal resistance moving in a rare medium. The synthesis of optimal solutions is described. A…

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal…

A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid…

We discuss and compare several geometric structures which imply an upper bound to the acceleration of a particle measured in its rest system. While all of them have the same implications on the motion of a point particle, they differ in…

We analyse a mechanical system in two-dimensional relative motion with friction. Although the system is simple, the peculiar interplay between two kinetic friction forces and gravity leads to the wide range of admissible solutions exceeding…

The dynamical friction force experienced by a body moving at relativistic speed in a gaseous medium is examined. This force, which arises due to the gravitational interaction of the body with its own gravitationally-induced wake, is…

Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in…

For a given $\lambda >0$, a convex body in $\mathbb R^n$ is $\lambda$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/\lambda$. In this note, we show that among all $\lambda$-convex bodies in $\mathbb…

In this work, we extend the convex bodies chasing problem (CBC) to an adversarial setting, where an agent (the Player) is tasked with chasing a sequence of convex bodies generated adversarially by another agent (the Opponent). The Player…

We consider the following problem: minimize the functional $\int_\Omega f(\nabla u(x))\, dx$ in the class of concave functions $u: \Omega \to [0,M]$, where $\Omega \subset \mathbb{R}^2$ is a convex body and $M > 0$. If $f(x) = 1/(1 +…

We consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid…

We look for particular solutions to the restricted three-body problem where the bodies are allowed to either lose or gain mass to or from a static atmosphere. In the case that all the masses are proportional to the same function of time,we…

We consider a body in a parallel flow of non-interacting particles. The interaction of particles with the body is perfectly elastic. We introduce the notions of a body of zero resistance, a body that leaves no trace, and an invisible body,…

Let $K$ be a convex body (a non-empty compact convex set) in $n$-dimensional Euclidean space. A set $B$ is called a barrier (or an `opaque set') for $K$ if every line that intersects $K$, also intersects $B$. Although this concept was…

A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid…

For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The thickness…

The paper present a model for the drag force between a resistive medium and a solid body using the hypothesis that the drag force is created by the adhesion of some particles of the resistive medium on the solid body's surface. The study…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

We consider the motion of a rigid body, governed by the Navier-Stokes equations in a bounded domain. Navier's condition is prescribed on the boundary of the body. We give the global in a time solvability result of weak solution. The result…