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The Extended Monotropic Programming problem deals with a more general extension to all n variables at a time than the two variables at a time extension you are interested in, except that the literature I have seen deals only with the constraint set $C$ being a closed linear subspace, not a general convex set. Specifically: $\min_{x \in \text{closed linear ... 0 The classical material derivative$D\varphi/Dt$of a test function$\varphi \in C_c^\infty(\mathbb{R}_+\times D)$is obtained by setting $$\dfrac{D\varphi}{Dt}(x) := \dfrac{\partial\tilde\varphi}{\partial t}(0,x)\;\;\mbox{with}\;\; \tilde\varphi(t,x) = \varphi(t,T_t(x))$$ for$x\in D$. Expanding out using the chain rule, we have $$\dfrac{D\varphi}{Dt}(x) =... 2 Brownian motion, i.e. Wiener measure, is a good source of ideas and examples here. For instance if W_t is 1-dimensional standard Brownian motion at time t and$$P(\forall x\,F(x)=x^2)=1$$and Y=W_1 then F and Y are independent but$$E(F(Y))=1\ne E(F(E(Y)))=E(F(0))=0.$$(Actually here we just need that W_1 is a standard normal random variable.) 2 The answer is yes, at least in the limiting case where the number of points tends to infinity. Specifically, this is known as the quantizer problem (see Chapter 2 of Sphere Packings, Lattices and Groups by Conway and Sloane). The two-dimensional version of the problem was solved by Fejes Tóth, who showed that the hexagonal lattice is optimal. László Fejes ... 2 We have 0\le a\le b\le1 and T\in(0,\infty). We want to know when there is a positive constant C such that$$\int_0^T dt\, \int_a^b dx\, u^2(x-t)\geq C\int_0^1 dx\,u^2(x) \tag{1}$$for all measurable functions$u\colon\mathbb R\to\mathbb R$such that$u(x)=0$for$x\notin(a,b)$. The answer is: never. Indeed, without loss of generality$a<b\$. The left-...