Study on minimizing perimeter in unbounded convex bodies without boundary regularity.
problem Minimizing perimeter under volume constraint in unbounded convex bodies.
method Introducing uniform geometry, asymptotic cylinders, approximation, and approximation argument.
result Existence of isoperimetric regions in generalized sense and strict concavity of isoperimetric profile.
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric pr…
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of Rn which are homeomorphic to Rn−1. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonic…
We introduce a particular class of unbounded closed convex sets of Rd+1, called F-convex sets (F stands for future). To define them, we use the Minkowski bilinear form of signature (+,...,+,−) instead of the usual scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic space $\H^d$. Impo…
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.
New index characterizes non-smooth Zoll convex bodies.
problem Characterizing non-smooth Zoll convex bodies.
method Defining systolic S1-index and using it to introduce generalized Zoll convex bodies. result Generalized Zoll convex bodies coincide with classical ones under certain conditions.
Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.
problem Characterizing quadrics among affine hyperspheres based on section centroid collinearity.
method Extending Meyer and Reisner's theorem to unbounded convex sets and identifying additional assumptions.
result Ellipsoids, paraboloids, and one sheet of a two-sheeted hyperboloid are the only quadrics satisfying the centroid collinearity condition.
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…
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 Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
Study of smooth convex bodies up to congruence.
problem Understanding hyperspaces of smooth convex bodies up to congruence.
method Systematic study of hyperspaces of convex bodies, focusing on C∞ and C1 smoothness, and using homeomorphism and congruence concepts. result Determine the homeomorphism type of positively curved C∞ convex bodies and their quotient by isometries. Solves Alexandrov's problem for hyperbolic convex bodies.
problem Finding a convex body with a given curvature measure in hyperbolic space.
method Defined Gauss curvature measure, proved existence and uniqueness of solution.
result Uniqueness of the solution to Alexandrov's problem in hyperbolic space.
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 n-dimensional convex bodies for n≥2. We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.
Strongly convex bodies can be approximated by smooth ones.
problem Approximating strongly convex bodies with smooth ones.
method Using C2 locally strongly convex bodies. result Smooth approximations of strongly convex bodies exist and can be controlled in terms of Hausdorff distance.
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…
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 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.
Study on Lp affine surface areas and their inequalities for convex bodies.
problem Understanding weighted Lp affine surface areas in convex bodies. method Investigating valuations, isoperimetric inequalities, and connections to f divergences. result Established isoperimetric inequalities for weighted Lp affine surface areas. The paper proves a theorem linking convex body centroids and category theory.
problem Understanding centroids of sections of convex bodies.
method Lusternik-Schnirelmann category theory.
result At least n hyperplanes exist such that the center of mass of their intersection with a convex body lies on the boundary of the convex body.
Established a new inequality for convex bodies in high dimensions.
problem Bounding the product of dual quermassintegrals of convex bodies and their polar sets.
method Induction on dimensions, focusing on convex bodies in \(\mathbb{R}^{n+1}\).
result Proved a generalised Blaschke-Santalò inequality with upper bounds.
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 λ-sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies. result The thick λ-sausage body is the unique minimizer of area among all bodies with a given length and curvature constraints. Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
The study confirms two cases of the convex body isoperimetric conjecture in the plane.
problem The least perimeter to enclose a given area inside a unit disk is greater than inside any other convex set.
method Examined symmetric domains and perturbations of the unit disk.
result Two cases of the convex body isoperimetric conjecture are confirmed.
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 (1,1)-classes can be realized by convex bodies. Sharp stability results for reverse isoperimetric inequalities in 2D.
problem Reverse isoperimetric inequalities in the plane.
method Stability analysis of λ-convex bodies and convex bodies with smooth boundaries. result Sharp stability results for reverse isoperimetric inequalities, including inradius and Cheeger inequalities.
The paper studies properties of Orlicz-Petty bodies and related geometric areas.
problem Properties of Orlicz-Petty bodies and related geometric areas.
method Established properties through the existence and uniform boundedness of Orlicz-Petty bodies.
result Geominimal surface areas are continuous under certain conditions on convex bodies.
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.
This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
The study connects the average number of solutions to mixed volumes of convex bodies.
problem Finding a relationship between the average number of solutions to systems of equations and mixed volumes of convex bodies.
method Developed Banach metrics in vector spaces, constructed Banach convex bodies in the cotangent bundle of X, and calculated the average number of solutions as the mixed symplectic volume of these bodies. result The average number of solutions is equal to the mixed symplectic volume of Banach convex bodies.
The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.
problem Finding the smallest volume among λ-convex bodies of a given surface area. method Using λ-convex bodies and analyzing their properties in model spaces of constant curvature. result The λ-convex lens is the unique minimizer of volume among all λ-convex bodies of given surface area in R3. We study relations of some classes of k-convex, k-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{k-circular convex} and \textrm{k-circular visible} ones. Investigati…
The paper explores volume product and slicing conjectures using convex body deformations.
problem Volume product and slicing conjectures in convex geometry.
method Study of variational aspects of volume product functional under projective deformations.
result Provides a proof of a theorem by Klartag and identifies critical convex bodies.
The paper confirms a conjecture about convex bodies and their properties.
problem The conjecture about the mean width of convex bodies.
method Analyzing λ-convex bodies and their intersections of balls. result The λ-convex lens maximizes the mean width among all λ-convex bodies with a given inradius. 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 C1 norms, dual Santaló point for smooth curved unit balls. result Existence and uniqueness of Santaló point for convex bodies in normed spaces.
The paper explores centroids and static equilibrium points in non-Euclidean geometries.
problem Investigating centroids and static equilibrium points in spherical, hyperbolic, and normed spaces.
method Extending Gal'perin's work, the paper examines convex bodies in these spaces and analyzes the minimum number of equilibrium points.
result Every plane convex body in any of these spaces has at least four equilibrium points, and there are mono-monostatic convex bodies in 3D spherical, hyperbolic, and certain normed spaces.
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…
Formula calculates equilibria of convex bodies based on evolute winding number.
problem Determining equilibria of convex bodies in terms of their geometric properties.
method Formula derived from winding number of evolute of convex body boundary.
result Formula extends to cases where center of mass lies on evolute.
The Weyl problem is extended to hyperbolic and anti-de Sitter spaces, connecting geometry, analysis, and group theory.
problem The classical Weyl problem for surfaces in hyperbolic and anti-de Sitter spaces.
method Generalizations of the Weyl problem to unbounded convex subsets and convex surfaces, focusing on thin and thick asymptotic boundaries.
result Connections to Kleinian groups, complex analysis, circle packings, and grafting on the hyperbolic disk.
New theorem proves convex bodies with specific curvature measures are rescaled Wulff shapes.
problem Characterizing convex bodies based on anisotropic curvature measures.
method Analyzing k-th anisotropic curvature measures and their relation to anisotropic perimeter.
result Arbitrary convex bodies with specific curvature measures are rescaled Wulff shapes.
Study utility maximization in financial markets with bounded and unbounded payoffs.
problem Utility maximization in financial markets with constraints and unbounded payoffs.
method Combines quadratic backward stochastic differential equations and convex duality.
result Established utility indifference valuation, regime switching, and consumption-investment problems in unbounded markets.
The paper examines the failure of Brunn-Minkowski inequality for certain convex bodies.
problem Brunn-Minkowski inequality for q-th dual quermassintegrals with q>n. method Second variation argument, dimension reduction, Hadwiger's inequality, singular weighted Reilly formula, coordinate-slice Hardy inequality.
result Established the inequality for unconditional convex bodies in the full range 0<q≤n+1. Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k-th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.
problem Proving the log-Brunn-Minkowski inequality for convex bodies and zonoids.
method Establishing monotonicity of the deficit in the LLBM under line segment addition.
result Equality in LLBM for smooth convex bodies occurs only for homothetic bodies.
In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in R3 with constant width, constant brightness, and boundary of class C2 is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.
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…