Your notation is quite strange to me, and very non-standard, at least coming from differential geometry.

You go though some length to define the interior of the $n$-simplex $$\mathrm{int}(\Delta^n) := \bigl\{ x \in \mathbb{R}^{n+1} \mid \sum_{i=0}^n \xi = 1 ~~\text{and} ~~ x_i > 0 ~~\text{for all}~~ i\bigr\}.$$ This is an $n$-dimensional submanifold of $\mathbb{R}^{n+1}$ and it is flat with respect to the induced metric.

However, apparently you want to define a non-standard metric on this manifold. This is where things get incomprehensible for me. First of all, what is $\xi$? You write it is "the" global coordinate system. Obviously there are various coordinate symstems on $\mathrm{int}(\Delta^n)$ (e.g. those given by one of the projection maps along one of the axes), but I do not see how there would be a canonical one.

Secondly, what is $p_\xi$? You write that $\mathcal{P} = \{p_\xi\}$, but I do not see how to interpret this as a meaningful set. Then again, $p_\xi$ seems to be a scalar, because you take its $\log$ later on.

Thirdly, the formula for Christoffel symbols generally looks different from the one you write. Probably it is right in this context, but as I said, I have no idea what $p_\xi$ should be.

And even then, you define $l_t = \log \dot{\gamma}(t)$, which does not make sense, as $\gamma$ is a path in a manifold, where you cannot take the log of. Even if you identify $\dot{\gamma}(t)$ with the corresponding coordinates in you coordinate system, that is still a vector in $\mathbb{R}^n$, where I do not know how to take the $\log$ of.

So, quite frankly, the reason that you didn't get a response on math.StackExchange might be that you do not define a single thing you write.

I do not see why your metric is a sensible object, because it depends highly on your choice of coordinates (I am not even sure that your formula defines a metric, but I may be overlooking something). Furthermore, it is not clear at all why your Christoffel symbols are well-defined, as they have to transform in a certain way under coordinate change to actually give a connection.

However, if we assume that they actually do define a connection, you just have to check the definition: A path $\gamma$ is a geodesic if it fulfills the geodesic equation $$\frac{\nabla}{d t} \dot{\gamma}(t) \equiv 0,$$ where in terms of Christoffel symbols, $$\frac{\nabla}{d t} \dot{\gamma}(t) = \ddot{\gamma}(t) + \Gamma_{ij}^k \,\dot{\gamma}(t)\,\dot{\gamma}(t).$$ If the formula from the paper is true in your situation, you can just verify it by explicitly computing every term according to your definitions.

However, there should be an invariant (coordinate-independent) definition of the connection, which should give way more insight in what this actually means (and shows that everything is indeed well-defined).

Earlier Post:

Your notation is quite strange to me, and very non-standard, at least coming from differential geometry.

You go though some length to define the interior of the $n$-simplex $$\mathrm{int}(\Delta^n) := \bigl\{ x \in \mathbb{R}^{n+1} \mid \sum_{i=0}^n \xi = 1 ~~\text{and} ~~ x_i > 0 ~~\text{for all}~~ i\bigr\}.$$ This is an $n$-dimensional submanifold of $\mathbb{R}^{n+1}$ and it is flat with respect to the induced metric.

However, apparently you want to define a non-standard metric on this manifold. This is where things get incomprehensible for me. First of all, what is $\xi$? You write it is "the" global coordinate system. Obviously there are various coordinate symstems on $\mathrm{int}(\Delta^n)$ (e.g. those given by one of the projection maps along one of the axes), but I do not see how there would be a canonical one.

Secondly, what is $p_\xi$? You write that $\mathcal{P} = \{p_\xi\}$, but I do not see how to interpret this as a meaningful set. Then again, $p_\xi$ seems to be a scalar, because you take its $\log$ later on.

Thirdly, the formula for Christoffel symbols generally looks different from the one you write. Probably it is right in this context, but as I said, I have no idea what $p_\xi$ should be.

And even then, you define $l_t = \log \dot{\gamma}(t)$, which does not make sense, as $\gamma$ is a path in a manifold, where you cannot take the log of. Even if you identify $\dot{\gamma}(t)$ with the corresponding coordinates in you coordinate system, that is still a vector in $\mathbb{R}^n$, where I do not know how to take the $\log$ of.

So, quite frankly, the reason that you didn't get a response on math.StackExchange might be that you do not define a single thing you write.