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Let $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$ and $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ be given and satisfy $$\sum_{i=1}^m \alpha_i =1 = \sum_{j=1}^n\beta_j.$$ Define $\P …

Let $\mathcal P$ be the set of probability distributions on $\mathbb R$ of finite first order, i.e. $\mu\in\mathcal P$ if $$\int_{\mathbb R} |t|\mu(dt)<\infty.$$ For $\mu\in\mathcal P$, define its put …

We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the populatio …

This is a continuation of Should coffee machines be deconcentrated? Recall that some region is denoted by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the peopl …

For $p\ge 1$, let $\mathcal P_p(\mathbb R)$ be the set of probability measures on $\mathbb R$ of finite $p^{\rm th}$ moment. Denote by $W_p$ the Wasserstein metric of order $p$ and by $\preceq$ the in …

Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is symm …