SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM

Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz’s square peg problem. We prove Hadwiger’s 1971 conjecture that any simple loop in $3$ -space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop. If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces which can be rearranged to form two loops of equal length. (The previous two results are independently due to Schwartz.) A rectifiable loop in $d$ -space can be cut into $(r-1)(d+1)+1$ pieces that can be rearranged by translations to form $r$ loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain inscribability results for the class of all continuous curves.

From Biology to Physics and Back: The Problem of Brownian Movement

This article focuses on the history of theoretical ideas but also on the developments of experimental tools. The experiments in our laboratory are used to illustrate the various developments associated with Brownian movement. In the first part of this review, we give an overview of the theory. We insist on the pre-Einstein approach to the problem by Lord Rayleigh, Bachelier, and Smoluchowski. In the second part, we detail the achievements of Perrin, measuring Avogadro's number, quantifying the experimental observations of Brownian movement, and introducing the problem of continuous curves without tangent, a precursor to fractal theory. The third part deals with modern application of Brownian movement, escape from a fixed optical trap, particle dynamics on a moving trap, and finally development of Brownian thermal ratchets. Finally, we give a short overview of bacteria motion, presented like an active Brownian movement with very high effective temperature.

Tangent Lines and Lipschitz Differentiability Spaces

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces.We show that any tangent space of a Lipschitz differentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space.