This is a talk I will give on Monday, December 15th, 2025 at the Seminar on Domain Theory, hosted by the TianYuan Mathematics Center in Kunming, Yunnan, PRC. The organizers are Hui Kou (寇辉, Sichuan University, Chengdu, Sichuan, PRC), Xiaodong Jia (贾晓东, Hunan University, Changsha, Hunan, PRC), and myself.
Abstract
I will give an introduction to barycentric algebras, the plain, the preordered and the (semi)topological. Those include all convex subsets of (plain, preordered, semitopological) cones, but there are a few more. Then I will turn to pointed barycentric algebras, the preordered and the (semi)topological. The leading examples are spaces of probability, and subprobability valuations (or measures).
Slides
The full slides, with animation steps; the shorter presentation, without them.
Videos
- Cones: what we know about them (9:48)
- Barycentric algebras: the plain, the preordered and the (semi)topological (24:32)
- Pointed barycentric algebras: the preordered and the (semi)topological (19:02)
Updates
I have left a few questions open in the slides, and here are a few updates on these questions.
- (February 03rd, 2026) Question 3 is now solved, but leads to more questions.
- On December 17th, 2025, I found a topological barycentric algebras that is not locally convex, even not weakly locally convex. This is inspired by Tychonoff’s example of a non-locally convex topological vector space, lp(N), with 0<p<1. Roughly, we take the subspace of sequences with non-positive values, and give it a quasi-metric that is defined just like the lp metric, except that we replace |x—y| by max(x—y, 0) everywhere.
This non-locally convex topological barycentric algebra is not embeddable. Its free semitopological cone, then, is locally convex. - On February 02nd, 2026, I found a semitopological cone that is not locally convex, even not weakly locally convex. (On February 18th, 2026, I found a bug in the argument, and the following includes the fix.) This is very artificial. You take the set of all maps from [0, 1[ to R+, ordered pointwise, and you give it the finest topology coarser than the Scott topology and making + and . separately continuous such that 2χUn converges to the constant function 1, where Un is the collection of numbers that have a 0 as their n+1st bit when written in binary. Then you can show that fn ≝ (1/n) ∑i=n 2n–1 2χUn does not converge to 1: the set C ≝ ↓{fn | n ∈ N–{0}} ∪ ↓{f∞}, where f∞ ≝ limsup fn, is closed but does not contain 1. Now the complement of C is an open neighborhood of 1 that does not contain any convex neighborhood A of 1: otherwise, U ≝ int(A) would be an open neighborhood of 1, hence would contain 2χUn for n large enough; then A would contain the convex combination fn for n large enough, and that is impossible since A is disjoint from C, and C contains every fn.
This cone is quite probably not topological, only semitopological. Hence the remaining question is: can you find an example of a topological cone that is not (weakly) locally convex?
Also, I still do not know whether the notions ‘weakly locally convex’, ‘locally convex’ and ‘locally linear’ are distinct.
- On December 17th, 2025, I found a topological barycentric algebras that is not locally convex, even not weakly locally convex. This is inspired by Tychonoff’s example of a non-locally convex topological vector space, lp(N), with 0<p<1. Roughly, we take the subspace of sequences with non-positive values, and give it a quasi-metric that is defined just like the lp metric, except that we replace |x—y| by max(x—y, 0) everywhere.
