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Convex functions
Convex functions and convex sets are somewhat interdependent. When studying convex sets, some convex functions play a very important role (mainly the...
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From valuations on convex bodies to convex functions
A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by...
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Mean Convex Smoothing of Mean Convex Cones
We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space,...
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Convex Non-convex Variational Models
An important class of computational techniques to solve inverse problems in image processing relies on a variational approach: the optimal output is...
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Convex Optimization
Convex optimization or convex programming refers to the problem of minimizing convex functions over convex sets. Observe that we have been careful to...
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Convex sets
Convexity is a very intuitive and geometric notion, but plays a fundamental role in many (also abstract) branches of mathematics. In vector spaces,...
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Convex Functions
This chapter is devoted to convex functions, the rock star of optimization theory. In this section, we recall their key properties that matter for...
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Affine convex geometry – Part 1
This chapter is dedicated to the affine geometry of convex bodies (in view of their affine positions). Roughly speaking, we want to discuss how...
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Convex Sets
Recall that a convex set \(C \subset \mathcal {R}^d\) is a...
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Affine convex geometry – Part 2
The main objective of this chapter is to study some important (starlike or convex) bodies associated to a given convex body. These are the so-called...
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Groups of convex bodies
In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen’s polytope...
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Locally Convex Spaces
This chapter starts recalling the definitions of Hausdorff topological space, metric space, and normed space. Examples of Banach sequence spaces, of...
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Some further results on pointfree convex geometry
Inspired by locale theory, pointfree convex geometry was first proposed and studied by Yoshihiro Maruyama. In this paper, we shall continue to his...
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Lagrange Multipliers in Locally Convex Spaces
We give a general Lagrange multiplier rule for mathematical programming problems in a Hausdorff locally convex space. We consider infinitely many...
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Convex Analysis on Hadamard Spaces and Scaling Problems
In this paper, we address the bounded/unbounded determination of geodesically convex optimization on Hadamard spaces. In Euclidean convex...
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Best Möbius Approximations of Convex and Concave Map**s
We study the best Möbius approximations (BMA) to convex and concave conformal map**s of the disk, including the special case of map**s onto...
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Remarks on Wright-convex functions
In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles...
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Scalar curvature rigidity of convex polytopes
We prove a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary....
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On a weighted Hermite–Hadamard inequality involving convex functional arguments
In this paper, we are interested in investigating a weighted variant of Hermite–Hadamard type inequalities involving convex functionals. The approach...
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Stability of convex disks
We prove that topological disks with positive curvature and strictly convex boundary of large length are close to round spherical caps of constant...