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This follows from the Lemma. If $F$ is a face of $conv(X)$, $x\in F$ and a finite subset $A\subset X$ is inclusion-minimal subset for which $x\in conv(A)$, then $A\subset F$. Proof. Induction in $|A …

Let $Y$ be the span of $X$, $C=Y\cap (-\infty,0]^n$. Since $X$ and $C$ are disjoint convex sets in $Y$, there exists a non-zero functional $\eta\in Y^*$ which separates (not strictly) $X$ and $C$: $\e …

Yes, this is true. There are three cases. 1) $G_\varepsilon(x)=G(x+\theta)$ for some $\theta\in (-\varepsilon,\varepsilon)$. Then $G$ has a local, thus global, maximum at $x+\theta$, so we may take $ …

Yes, this is called 'majorization' or 'second order stochastic dominance' of measures (first term is used in analysis, second in probability). The idea is very simple: we partition the measure $\mu$ o …

$$y_i=\sum_j a_{ij} x_j\geqslant \prod_j x_j^{a_{ij} }$$ by Jensen inequality for logarithm. Now take the product over $i=1,2,\dots,n$.

Imagine that $v$ is (almost) equal to $(1,0,\ldots,0)$, then for $x=(x_1,\ldots,x_n)$, $\sum x_i=:na$, we should check that $(x_1-a,x_2-a,\ldots,x_n-a)\cdot(0,x_2,\ldots,x_n)\geqslant 0$, in other wor …

Yes. At first, if $h=\min(h_A,h_B)$ is convex (note that it is also 1-homogeneous), it is a support function of the body $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. Nex …

Let $A_0\subset A$ be the set where $\ell$ attains a minimum on $A$. It is a face of some dimension $k<d$. If $k=d-1$, we are done. Assume that $k<d-1$. Without loss of generality, $0\in A_0$ and more …

Is not it obvious (unlike Prékopa-Leindler)? We are given that for all $x_1,x_2$ we have $(f_1/g_1)^2 (x_1)\leqslant (g_2/f_2)^3(x_2)$, thus there exists $c>0$ such that $(f_1/g_1)^2 (x_1)\leqslant c^ …

Consider a 3d-rotation with respect to the axis generated by the unit vector $(a,b,c)$ to the angle $\theta$. Its matrix is $$ M=\pmatrix{\cos \theta+a^2(1-\cos\theta)&ab(1-\cos\theta)-c\sin\theta& ac …

Assume that $q\in C_2\setminus C_1$. Let $p$ be an interior point of $C_1$. Then the interval $(p,q)$ contains a boundary point of $C_1$ but only interior points of $C_2$. A contradiction.

Now I think that no, even if we remove the negative summands $-x_i+(1-x_i)\log(1-x_i)$. Note that our inequality becomes homogeneous, so we may forget that $x_i$ are less than 1. Choose $x_i=1+t_i$ so …

This subject (and its history) was discussed in Anosov's note in Математическое просвещение, http://www.mathnet.ru/links/a59beea5836a0d54828088c860feecf5/mp86.pdf