Study on discrete Okounkov bodies and their applications.
problem Understanding stability and thresholds in higher dimensions.
method Analysis of discrete Okounkov bodies and gap phenomena.
result Asymptotic analysis of stability and thresholds.
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48 results for “Okounkov bodies”
Research explores flat subspaces in complex projective manifolds using Okounkov bodies.
problem Existence of flat subspaces in complex projective manifolds.
method Utilizes the generalised Legendre transform to the Okounkov body and a result by Schwer--Lytchak.
result Sufficient conditions for the existence of flat subspaces are identified.
Study shows volumes of complex classes can be represented by convex bodies.
problem Understanding volumes of complex classes on Kähler manifolds.
method Approximation by partial Okounkov bodies, restricted volume properties, and bimeromorphic behavior of currents.
result Volume of transcendental big -classes can be realized by convex bodies.
The study constructs universal invariants for non-Archimedean metrics on projective varieties.
problem Understanding the singularity of non-Archimedean metrics on projective varieties.
method Constructing partial Okounkov bodies and Duistermaat--Heckman measures for non-Archimedean metrics.
result Generalization of Duistermaat--Heckman measures to finite energy metrics on Berkovich analytifications.
Optimal volume limit found for Kähler manifolds with positive Ricci curvature.
problem Bounding the volume of Kähler manifolds with positive Ricci curvature.
method Using δ-invariants and Newton--Okounkov bodies.
result Derive the optimal volume upper bound and new characterization of the complex projective space.
Constructs symplectic structures from rational functions on fans.
problem Creating symplectic structures from rational functions on fans.
method Constructs exact symplectic structures and polyhedral Hamiltonians.
result Level sets of polyhedral Hamiltonians are hypersurfaces of contact type.
We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …
Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …
We prove the quasimodularity of generating functions for counting torus covers, with and without Siegel-Veech weight. Our proof is based on analyzing decompositions of flat surfaces into horizontal cylinders. The quasimodularity arise as contour integral of quasi-elliptic functions. It provides an alternative proof of …
The abstract presents a new theorem using Ross-Witt Nyström correspondence and Berndtsson's theorem.
problem The abstract tackles the Ohsawa-Takegoshi extension theorem and its applications.
method The approach uses Ross-Witt Nyström correspondence and Berndtsson's theorem in \(\mathbb{C}^*\)-degeneration.
result The approach provides a quick proof of the Ohsawa-Takegoshi extension theorem without limits or singular weights.
We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity…
We show Vector Autoregressive Moving Average models with scalar Moving Average components could be estimated by generalized least square (GLS) for each fixed moving average polynomial. The conditional variance of the GLS model is the concentrated covariant matrix of the moving average process. Under GLS the likelihood …
Develops quantum circuits for faster learning with symmetry considerations.
problem Speeding up learning quantum states with symmetry considerations.
method Utilizes Okounkov-Vershik approach and Young-Jucys-Murphy elements to develop -equivariant convolutional quantum circuits.
result Proves -CQA generates any unitary in any given irrep sector, universal for SU() symmetry.
Proves log-concavity of cluster algebra coefficients for type .
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type .
result Proved log-concavity of coefficients for cluster algebra variables of type .
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
The paper proves a conjecture about the shape of floating bodies.
problem The shape of bodies of flotation and buoyancy.
method Modern differential geometry techniques.
result If a body of flotation is homothetic to a body of buoyancy, it must be an ellipse.
New index characterizes non-smooth Zoll convex bodies.
problem Characterizing non-smooth Zoll convex bodies.
method Defining systolic -index and using it to introduce generalized Zoll convex bodies.
result Generalized Zoll convex bodies coincide with classical ones under certain conditions.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing relative surface areas, proving invariance and inequalities, and using geometric interpretations.
result Established inequalities and a new notion of entropy for ball-convex bodies.
A groupoid called material groupoid is naturally associated to any simple body . The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid. Thus, the inclusion of these new objects in the theory of material bodies opens th…
The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.
problem Finding the unique minimizer of area for hyperbolic bodies with curvature constraints.
method Introduced the concept of 'thick -sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies.
result The thick -sausage body is the unique minimizer of area among all bodies with a given length and curvature constraints.
Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.
problem Defining surface area in non-Euclidean geometries.
method Generalizes illumination bodies to Riemannian spaces of constant curvature and projective Finsler geometries, proving their volume derivative defines surface area.
result Derivative of volume of illumination bodies defines surface area in non-Euclidean geometries.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…
We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, …
Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
problem Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
method Define the -illumination body and prove a generalization of Werner's formula.
result Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
The Orlicz-Petty bodiesmath.MG
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
The paper proves no multiple equichordal points exist in convex bodies.
problem Existence of multiple equichordal points in convex bodies.
method Topological tools like the Borsuk-Ulam theorem and analysis of convex body properties.
result Nonexistence of multiple equichordal points in -dimensional convex bodies for .
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…
When is a closed, orientable surface with genus , we show that the automorphism group of the compression body graph is the mapping class group. Here, vertices are compression bodies with exterior boundary , and edges connect pairs of compression bodies where one contains the other.
The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…
The paper extends inequalities for projection bodies to arbitrary measures.
problem Sharp bounds for volume ratios of convex bodies and their projection bodies.
method Generalizations of Zhang's inequality to arbitrary measures and extensions of the projection body operator.
result New Zhang-type inequalities for arbitrary measures and functions.
Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spacesmath.DG
The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.
problem Finding minimal fillings of convex bodies.
method Analyzing integral current spaces and proving rigidity properties.
result Convex bodies are the unique minimal fillings of their boundary metrics among integral current spaces and enjoy Lipschitz-volume rigidity.
In this paper, we are concerned with the 2D and 3D geometric shape generation by prescribing a set of characteristic values of a specific geometric body. One of the major motivations of our study is the 3D human body generation in various applications. We develop a novel method that can generate the desired body with c…
Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler…
Efficiently samples arbitrary compact bodies with polynomial complexity.
problem Uniform sampling from arbitrary compact bodies efficiently.
method Warm start algorithm under isoperimetry and volume growth condition.
result Substantial generalization of known results for convex and star-shaped bodies.
Study on affine surface areas and their inequalities for convex bodies.
problem Understanding weighted affine surface areas in convex bodies.
method Investigating valuations, isoperimetric inequalities, and connections to divergences.
result Established isoperimetric inequalities for weighted affine surface areas.
Efficient algorithm for sampling from arbitrary compact bodies.
problem Sampling from arbitrary compact bodies efficiently.
method Warm start algorithm with polynomial complexity.
result Substantial generalization of known results for convex and star-shaped bodies.
New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.
problem Periodic solutions of the 2n-body problem and their braid types.
method Analyzing braid types and stretch factors associated with pseudo-Anosov braids.
result Braids from new periodic solutions are of pseudo-Anosov type with stretch factors as metallic ratios.
Study spherical convex bodies using -floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced -floating areas and curvature entropy for spherical convex bodies.
result Established isoperimetric inequalities and dual isoperimetric inequalities.
We consider the class of -concave bodies in ; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the co…
Analytic convex bodies' Poincaré series extended holomorphically.
problem Analytic continuation of Poincaré series for convex bodies.
method Analytic continuation of Laplace transforms, holomorphic functions, and resolvent of multiplication operators.
result Poincaré series continues holomorphically to a conical neighborhood of the right half-plane, removing countable cuts and points.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
In this article we pose the problem of existence and uniqueness of convex body for which the projection curvature radius function coincides with given function. We find a necessary and sufficient condition that ensures a positive answer to both questions and suggest an algorithm of construction of the body. Also we fin…
Study examines how body segments respond to random vibrations.
problem Understanding human body responses to random vibrations.
method 35 participants were tested with random noise signals. Multiple linear regression models were created to determine influential predictors of peak translational gains.
result Multiple predictors, including motion direction and body segment, significantly influence peak translational gains.
Study on materials with disclinations, limiting their size.
problem Limiting the size of disclinations in materials with symmetries.
method Defining material-uniform hyperelastic bodies with disclinations, rigorously analyzing their properties.
result The size of disclinations is limited by the symmetries of the constitutive relation.
Study on Santaló point for convex bodies in normed spaces.
problem Exploring Santaló point for convex bodies in normed spaces.
method Existence and uniqueness proof for norms, dual Santaló point for smooth curved unit balls.
result Existence and uniqueness of Santaló point for convex bodies in normed spaces.