Related papers: Everything is possible: constructing spectrahedra …
Very recently Richter and Rogers proved that any convex geometry can be represented by a family of convex polygons in the plane. We shall generalize their construction and obtain a wide variety of convex shapes for representing convex…
A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron we are interested in the question of the smallest possible size $r$ of the matrices in the description by linear matrix inequalities. We show that for the…
Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear…
Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate…
We initiate the study of extremal problems about faces in convex rectilinear drawings of~$K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points…
We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual…
We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex…
A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary…
Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal…
Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene…
The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are…
In this work, we present a new efficient method for convex shape representation, which is regardless of the dimension of the concerned objects, using level-set approaches. Convexity prior is very useful for object completion in computer…
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…
A convex cone is said to be projectionally exposed (p-exposed) if every face arises as a projection of the original cone. It is known that, in dimension at most four, the intersection of two p-exposed cones is again p-exposed. In this paper…
Given n >= 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.
We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds…
We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex…
We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a…
We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…