DeepAI

# Formal Adventures in Convex and Conical Spaces

Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic axiomatization of convex sets, namely convex spaces, based on an operation taking barycenters of points. A convex space corresponds to a specific type that does not refer to a surrounding vector space. This simplifies the definitions of functions on it. We show applications including the convexity of information-theoretic functions defined over types of distributions. We also show how convex spaces are embedded in conical spaces, which are abstract real cones, and use the embedding as an effective device to ease calculations.

• 5 publications
• 4 publications
• 2 publications
07/02/2019

### Covering graphs with convex sets and partitioning graphs into convex sets

We present some complexity results concerning the problems of covering a...
02/19/2016

### First-order Methods for Geodesically Convex Optimization

Geodesic convexity generalizes the notion of (vector space) convexity to...
05/06/2022

### Convex Analysis at Infinity: An Introduction to Astral Space

Not all convex functions on ℝ^n have finite minimizers; some can only be...
01/29/2019

### Categorical Equivalences from State-Effect Adjunctions

From every pair of adjoint functors it is possible to produce a (possibl...
09/22/2020

### Strongly Convex Divergences

We consider a sub-class of the f-divergences satisfying a stronger conve...
05/04/2019

### A categorical construction for the computational definition of vector spaces

Lambda-S is an extension to first-order lambda calculus unifying two app...
11/02/2018

### Chasing Nested Convex Bodies Nearly Optimally

The convex body chasing problem, introduced by Friedman and Linial, is a...