Reconstruction of Centrally Symmetrical Convex Bodies by Projection Curvature Radii

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, and find applications to geometric tomography. We introduce the notion of a circular projection and investigate the problem of determination of $\eps$-convex bodi...

Generic 3D reconstruction from a single image is a difficult problem. A lot of data loss occurs in the projection. A domain based approach to reconstruction where we solve a smaller set of problems for a particular use case lead to greater returns. The project provides a way to automatically generate full 3-D renditions of actual symmetric images that have some prior information provided in the pipeline by a recognition algorithm. We provide a critical analysis on how this ca...

In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of lattice points in terms of volumes of suitable projections.

In this paper, we consider the problem: given a symmetric concave configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We show that there are some regions in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the union of these regions, it is always possible to choose positive masses to make the configuration central.

This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.

We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(\Pi K)/V^{d-1}(K)$. Namely, we prove that if $K$ is a 3-dimensional zonoid of volume 1, then its second projection body $\Pi^2K$ is contained in 8K, while if $K$ is any symmetric 3-dimensional convex body of volume 1, then $\Pi^2K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimension...

For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in $\R^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all $K_0$ with $\f K_0$ of class $C^2$ with positive radii of curvature). Assume that $K$ and $K_0$ have proportional 1st projecti...

In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a functional for certain value of the mass. We give a positive answer in dimension two while in higher dimension the situation is different. In fact, for small value of mass the ball centered at the origin is a local minimizer while for large...

Given a convex body $K\subset \mathbb R^2$ we say that a circle $\Omega\subset \text{int} \ K$ is an equipotential circle if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point $P=\Omega\cap AB$ it holds that $|AP|\cdot|PB|=\lambda$, for a suitable constant number $\lambda$. The main result in this article is the following: Let $K\subset\mathbb R^2$ be a convex body which has an equipotential circle $\mathcal B$ with centre $O$ in its inter...

We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the ellipsoid. Simple examples with non-unique minimal or maximal ellipsoids conclude this article.