Dual Curvature Research Articles

Chord measures in integral geometry and their Minkowski problems

AbstractTo the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.

Design of Metal–Organic Assemblies via Shape Complementarity and Conformational Constraints in Dual Curvature Ligands

Design of Metal–Organic Assemblies via Shape Complementarity and Conformational Constraints in Dual Curvature Ligands

On subspace concentration for dual curvature measures

We study subspace concentration of dual curvature measures of convex bodies K satisfying γ(−K)⊆K for some γ∈(0,1]. We present upper bounds on the subspace concentration depending on γ, which, in particular, retrieves the known results in the symmetric setting. The proof is based on a unified approach to prove necessary subspace concentration conditions via the divergence theorem.

Lp$L_p$‐cosine transform of the (p,q)${(p,q)}$‐th dual curvature measure

AbstractThe ‐th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the cosine transform of the ‐th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the Petty projection body and the centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known Petty projection inequality.

The 𝐿_{𝑝} Brunn-Minkowski inequalities for dual quermassintegrals

From a convex geometry viewpoint, we proved the L p L_p Brunn-Minkowski inequalities for q q -th dual quermassintegrals, when p ≥ q p\geq q . Based on these inequalities, we obtain relevant uniqueness results of the ( p , q ) (p,q) -th dual curvature measures (up to a dilation when p = q p=q ). As a special case q = 0 q=0 , we obtain the uniqueness of L p L_p integral curvature measure. Part of these uniqueness results were obtained before from different viewpoints.

The L p dual Minkowski problem about 0 < p < 1 and q > 0

Abstract The ( p , q p,q )-th dual curvature measures and the L p {L}_{p} dual Minkowski problem were recently introduced by Lutwak, Yang, and Zhang. In this paper, we give a solution to the existence part of the L p {L}_{p} dual Minkowski problem about 0 < p < 1 0\lt p\lt 1 and q > 0 q\gt 0 for arbitrary measures. This fills up previously obtained results.

Manifest electric-magnetic duality in linearized conformal gravity

We derive a manifestly duality-symmetric formulation of the action principle for conformal gravity linearized around Minkowski space-time. The analysis is performed in the Hamiltonian formulation, the fourth-order character of the equations of motion requiring the formal treatment of the three-dimensional metric perturbation and the extrinsic curvature as independent dynamical variables. The constraints are solved in terms of two symmetric potentials that are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. The action principle can be written in terms of these four dynamical variables, duality acting as simultaneous rotations in the respective spaces spanned by the three-dimensional metrics and the extrinsic curvatures. A twisted self-duality formulation of the equations of motion is also provided.

Dual curves associated with the Bonnet ruled surfaces

An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.

A note on the double dual graviton

The (free) graviton admits, in addition to the standard Pauli-Fierz description by means of a rank-two symmetric tensor, a description in which one dualizes the corresponding -curvature tensor on one column to get a -tensor, where D is the spacetime dimension. This tensor derives from a gauge field with mixed Yound symmetry called the ‘dual graviton’ field. The dual graviton field is related non-locally to the Pauli-Fierz field (even on-shell), in much the same way as a p -form potential and its dual -form potential are related in the theory of an abelian p -form. Since the Pauli-Fierz field has a Young tableau with two columns (of one box each), one can contemplate a double dual description in which one dualizes on both columns and not just on one. The double dual curvature is now a -tensor and derives from a gauge field with mixed Young symmetry, the ‘double dual graviton’ field. We show, however, that the double dual graviton field is algebraically and locally related to the original Pauli-Fierz field and, so, does not provide a truly new description of the graviton. From this point of view, it plays a very different role from the dual graviton field obtained through a single dualization. We also show that these equations can be obtained from a variational principle in which the variables to be varied in the action are (all) the components of the double-dual field as well as an auxiliary field with Young symmetry. By gauge fixing the shift symmetries of this action principle, one recovers the Pauli-Fierz action. Our approach differs from the interesting approach based on parent actions and covers only the free, sourceless theory. Similar results are argued to hold for higher spin gauge fields.

Single lumen, dual curvature tube for left bronchial intubation in pediatric patients

Single lumen, dual curvature tube for left bronchial intubation in pediatric patients

The dual Minkowski problem for symmetric convex bodies

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in Rn. A full solution to this is given when 1<q<n. The necessary and sufficient conditions turn out to be an explicit measure concentration condition. To obtain the results, a variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral which is obtained by using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

The Lp dual Minkowski problem for p > 1 and q > 0

General Lp dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. Lp dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the Lp dual Minkowski problem, which concerns the characterization of Lp dual curvature measures. In this paper, we solve the existence part of the Lp dual Minkowski problem for p>1 and q>0, and we also discuss the regularity of the solution.

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem I

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

Subspace concentration of dual curvature measures of symmetric convex bodies

We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body. © 2018 International Press of Boston, Inc. All Rights Reserved.

Helical CT Scan Evolution of Periprosthetic Cysts of a Stainless Steel DLC Coated Total Ankle Arthroplasty at Four Years Apart and Survival at 13 Years

Category: Ankle Introduction/Purpose: Total ankle arthroplasty (TAA) is the alternative to tibio-talar fusion in the non-conservative operative treatment of osteoarthritis. It preserves joint range of motion and prevents adjacent joints while improving quality of life and relief. Long-term studies frequently show the presence of periprosthetic cysts, which is linked to an increased risk of revision, pain or lameness. In 2012, a stainless steel, DLC coated, TAA was subjected to a clinical review and a CT scan analysis. The main objective of this study was to analyze periprosthetic cysts evolution (numbers and volumes) on helical CT scans at 4 years apart of this same cohort of patients. The secondary objective was to present clinical results and survival curve of the implant. Methods: We retrospectively reviewed the cohort of 68 cases from 2012. The average follow-up was 13 ± 6 years. 42 CT analysis of the Periprosthetic cysts, were performed at the last follow-up and compared to the 2012 analysis. Cysts were classified in 3 groups (A: 0-200mm2, B: 200-400mm2, C:&gt; 400 mm2) distributed over 7 areas. Numbers, locations and sizes were listed. The functional evaluation was performed in 2012 and at the last follow-up with the American Orthopedic Foot and Ankle Society Ankle and Hindfoot Score (AOFAS-AHS) and with the range of motion. TAR survival at 5 years were calculated for the 68 patients, and for the last 34 to avoid the learning curve effect. Two end points were defined: revision or removal of the implant for any reasons and revision or removal of the implant for aseptic loosening. TAR survival at 10 years was also analyzed thanks to the long follow-up. Results: Cysts size and prevalence were not significantly different at 4-year apart (p&gt; 0.05). No type C cysts were revealed. The mean AOFAS-AHS was 77.6 ± 15.4 [38-98] in 2012 and stable at 75 ± 18.2 [16-97] at the last follow-up. Pre-operative ankle range of motion and at the last follow-up were not significantly different. The 5-year survival for the last 34 cases were 85.2% [73.2; 97.2] for any cause of revision and 90.7% [80.7; 100] for aseptic loosening. The 10-year survival for the last 34 cases were 77.1% [67.3; 87.4] for any cause of revision and 85.4% [73.4; 98.4] for aseptic loosening. Between 2012 and 2016, one case was revised for aseptic loosening of the talar component and two cases needed fusion. Conclusion: This study shows no significant increase of cysts volume or number at 4-year apart with a stainless steel, DLC coated TAA. The clinical results and 5-year survival are similar to those in the literature. The strength of this study is its mean follow-up greater than 13 years which allowed a 10-year survival and a long-term CT scan analysis. The implant design (spherical tibial component, dual curvature PE insert, trochlear-shaped talar component), its materials (DLC coated stainless steel) and its fixation (alumina coating) ensure long-term similar clinical results, with no break after 5-year survival, and stability of cysts prevalence and size.

Relative-locality phenomenology on Snyder spacetime

We study the effects of relative locality dynamics in the case of the Snyder model. Several properties of this model differ from those of the widely studied models: for example, in the Snyder case the action of the Lorentz group is preserved, and the composition law of momenta is deformed by terms quadratic in the inverse Planck energy. From the investigation of time delay and dual curvature lensing we deduce that, because of these differences, in the Snyder case the properties of the detector are essential for the observation of relative locality effects. The deviations from special relativity do not depend on the energy of the particles and are much smaller than in the case, so are beyond the reach of present astrophysical experiments. However, these results have a conceptual interest, because they show that relative-locality effects can occur even if the action of the Lorentz group on phase space is not deformed.

On the Lp dual Minkowski problem

The Lp dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Lp Brunn–Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for Lp dual curvature measure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the Lp dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.

On the planar dual Minkowski problem

In this paper, we resolve the planar dual Minkowski problem, proposed by Huang et al. (2016) [31] for all positive indices without any symmetry assumption. More precisely, given any q>0, and function f on S1, bounded by two positive constants, we show that there exists a convex body Ω in the plane, containing the origin in its interior, whose dual curvature measure C˜q(Ω,⋅) has density f. In particular, if f is smooth, then ∂Ω is also smooth.

Lp dual curvature measures

A new family of geometric Borel measures on the unit sphere is introduced. Special cases include the Lp surface area measures (which extend the classical surface area measure of Aleksandrov and Fenchel & Jessen) and Lp-integral curvature (which extends Alkesandrov's integral curvature) in the Lp Brunn–Minkowski theory. It also includes the dual curvature measures (which are the duals of Federer's curvature measures) in the dual Brunn–Minkowski theory. This partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes.

Necessary subspace concentration conditions for the even dual Minkowski problem

We prove tight subspace concentration inequalities for the dual curvature measures C˜q(K,⋅) of an n-dimensional origin-symmetric convex body for q≥n+1. This supplements former results obtained in the range q≤n.