  If $\alpha$ is a probability on $\mathbb{R}^d$ and $t>0,$ consider the
Dirichlet random probability $P_t\sim\mathcal{D}(t\alpha) ;$ it is such that
for any measurable partition $(A_0,\ldots,A_k)$ of $\mathbb{R}^d$ then
$(P_t(A_0),\ldots,P_t(A_k))$ is Dirichlet distributed with parameters
$(t\alpha(A_0)\ldots,t\alpha(A_k)).$ If
$\int_{\mathbb{R}^d}\log(1+\|x\|)\alpha(dx)<\infty$ the random variable
$\int_{\mathbb{R}^d}xP_t(dx)$ of $\mathbb{R}^d$ does exist and we denote by
$\mu(t\alpha)$ its distribution. The Dirichlet curve associated to the
probability $\alpha$ is the map $t\mapsto \mu(t\alpha).$ It has simple
properties like $\lim_{t\searrow 0}\mu(t\alpha)=\alpha$ and $\lim_{t\rightarrow
\infty}\mu(t\alpha)=\delta_m$ when $m=\int_{\mathbb{R}^d} x\alpha(dx)$ exists.
The present paper shows first that if $m$ exists and if $\psi$ is a convex
function on $\mathbb{R}^d$ then $t\mapsto
\int_{\mathbb{R}^d}\psi(x)\mu(t\alpha)(dx)$ is a decreasing function, which
means that $t\mapsto \mu(t\alpha)$ is decreasing according to the Strassen
convex order of probabilities. The second aim of the paper is to prove a group
of results around the following question: if $\mu(t\alpha)=\mu(s\alpha)$ for
some $0\leq s<t$, can we claim that $\mu$ is Cauchy distributed in
$\mathbb{R}^d?$
