Mathematics

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek-Geiringer-Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 5 and its corresponding Algorithm 1 which decide if a configuration is epsilon-locally rigid, a notion we define. A configuration which is epsilon-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius epsilon in configuration space. Deciding epsilon-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.

We prove an equilibrium stressability criterium for trivalent multidimensional tensegrities. The criterium appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms.

We prove that, for any positive integer m , a segment may be partitioned into m possibly degenerate or empty segments with equal values of a continuous function f of a segment, assuming that f may take positive and negative values, but its value on degenerate or empty segments is zero.

We show that every graded nilpotent Lie group G of step r , equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading, (this includes all Carnot groups with Carnot-Caratheodory metric) is Markov p -convex for all p∈[2r,∞) . We also show that this is sharp whenever G is a Carnot group with r≤3 , a free Carnot group, or a jet space group; such groups are not Markov p -convex for any p∈(0,2r) . This continues a line of research started by Li who proved this sharp result when G is the Heisenberg group. As corollaries, we obtain new estimates on the non-biLipschitz embeddability of some finitely generated nilpotent groups into nilpotent Lie groups of lower step. Sharp estimates of this type are known when the domain is the Heisenberg group and the target is a uniformly convex Banach space or L 1 , but not when the target is a nonabelian nilpotent group.

We prove that every Jordan curve in R 2 inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.

Ketterer and Rajala showed an example of metric measure space, satisfying the measure contraction property MCP(0,3) , that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the CD(0,?? condition, proving the non-constancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of Lott-Sturm-Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allows to answer to some open question proposed by Schultz, about strict curvature dimension bounds and their stability with respect to the measured Gromov Hausdorff convergence.

Let $K \in \R^d$ be a convex body, and assume that L is a randomly rotated and shifted integer lattice. Let K L be the convex hull of the (random) points K∩L . The mean width W( K L ) of K L is investigated. The asymptotic order of the mean width difference W(łK)−W((łK ) L ) is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as ł→∞ .

In this paper geometry of Gromov-Hausdorff distance on the class of all metric spaces considered up to an isometry is investigated. For this class continuous curves and their lengths are defined, and it is shown that the Gromov-Hausdorff distance is intrinsic. Besides, metric segments are considered, i.e., the classes of points lying between two given ones, and an extension problem of such segments beyond their end-points is considered.

In this paper we study connections between Besov spaces of functions on a compact metric space Z , equipped with a doubling measure, and the Newton--Sobolev space of functions on a uniform domain X ε . This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of Z . To do so, we construct a family of hyperbolic fillings in the style of the work of Bonk and Kleiner and the work of Bourdon and Pajot. Then for each parameter β>0 we construct a lift μ β of the doubling measure ν on Z to X ε , and show that μ β is doubling and supports a 1 -Poincaré inequality. We then show that for each θ with 0<θ<1 and p≥1 there is a choice of β=p(1−θ)logα such that the Besov space B θ p,p (Z) is the trace space of the Newton--Sobolev space N 1,p ( X ε , μ β ) when ε=logα . Finally, we exploit the tools of potential theory on X ε to obtain fine properties of functions in B θ p,p (Z) , such as their quasicontinuity and quasieverywhere existence of L q -Lebesgue points with q= s ν p/( s ν −pθ) , where s ν is a doubling dimension associated with the measure ν on Z . Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov in R n .

We generalize the extension and trace results of Björn-Björn-Shanmugalingam \cite{BBS21} to the setting of complete noncompact doubling metric measure spaces and their uniformized hyperbolic fillings. This is done through a uniformization procedure introduced by the author that uniformizes a Gromov hyperbolic space using a Busemann function instead of the distance functions considered in the work of Bonk-Heinonen-Koskela \cite{BHK}. We deduce several corollaries for the Besov spaces that arise as trace spaces in this fashion, including the existence of representatives that are quasicontinuous with respect to the Besov capacity, the existence of L p -Lebesgue points quasieverywhere with respect to the Besov capacity, embeddings into Hölder spaces for appropriate exponents, and a stronger Lebesgue point result under an additional reverse doubling hypothesis on the measure. We also obtain several Poincaré-type inequalities relating integrals of Besov functions over balls to integrals of upper gradients of extension of these functions to a uniformized hyperbolic filling of the space.