Generalized matrix-fractional (GMF) functions are a class of matrix support func4 tions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix 5 optimization problems associated with inverse problems, regularization and learning. In this paper 6 we dramatically simplify the support function representation for GMF functions as well as the rep7 resentation o...

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime (M, gμν) or an initial data set (Σ, hij , Kij) admitting a suitably defined convex function. We show how the existence of a convex function on a spacetime places restrictions on the properties...

Let S be a finite set with n elements in a real linear space. Let JS be a set of n intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved that each homothet of conv S that is contained in aff S can be obtained using this operator. A vari...

If two symmetric convex bodies K and L both have nicely bounded sections, then the intersection of random rotations of K and L is also nicely bounded. For L being a subspace, this main result immediately yields the unexpected “existence vs. prevalence” phenomenon: If K has one nicely bounded section, then most sections of K are nicely bounded. The main result represents a new connection between...

– The Brunn-Minkowski theory is a central part of convex geometry. At its foundation lies the Minkowski addition of convex bodies which led to the definition of mixed volume of convex bodies and to various notions and inequalities in convex geometry. Its origins were in Minkowski’s joining his notion of mixed volumes with the Brunn-Minkowski inequality, which dated back to 1887. Since then it h...

A projective d-arrangement of n hyperplanes H(d, n) is a finite collection of hyperplanes in the real projective space Pd such that no point belongs to every hyperplane of H(d, n). Any arrangement H(d, n) decomposes Pd into a d-dimensional cell complex K . We may call cells of H(d, n) the d-cells of K , and facets of H(d, n) the (d − 1)-cells of K . Clearly any cell of H(d, n) has at least (res...

We introduce a unified algorithmic framework, called proximal-like incremental aggregated gradient (PLIAG) method, for minimizing the sum of smooth convex component functions and a proper closed convex regularization function that is possibly non-smooth and extendedvalued, with an additional abstract feasible set whose geometry can be captured by using the domain of a Legendre function. The PLI...

1 Abstract We present parallel computational geometry algorithms that are scalable, architecture independent, easy to implement, and have, with high probability, an optimal time complexity for uniformly distributed random input data. Our methods apply to multicomputers with arbitrary interconnection network or bus system. The following problems are studied in this paper: (1) lower envelope of l...

One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak or without constraint qualifications. The main idea is to work with intrinsic topologies which reflect some geom...