We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in ${ℝ}^n$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of ${ℝ}^n$ satisfying $d(Y\cap K,B_2^k)\le C(1+\surd (k/ln(n/\left(kln(n+1)\right)))$. This formulation of Dvoretzky’s theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.

We propose new alternative theorems for convex infinite systems which constitute the generalization of the corresponding to Gale, Farkas, Gordan and Motzkin. By means of these powerful results we establish new approaches to the Theory of Infinite Linear Inequality Systems, Perfect Duality, Semi-infinite Games and Optimality Theory for non-differentiable convex Semi-Infinite Programming Problem.

There is a constant c such that for every n ∈ ℕ, there is an Nₙ so that for every N≥ Nₙ there is a polytope P in ℝⁿ with N vertices and $volₙ\left(B₂ⁿ△P\right)\le cvolₙ\left(B₂ⁿ\right)N^{-2/(n-1)}$ where B₂ⁿ denotes the Euclidean unit ball of dimension n.

Let U, V be two symmetric convex bodies in ${ℝ}^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n\in U$ such that, for each choice of signs ${\epsilon }_1,...,{\epsilon }_n=\pm 1$, one has ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\notin rV$ where $r={\left(2\pi e^2\right)}^{-1/2}{n}^{1/2}\left(\right|U|/{\left|V\right|)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left(u_n\right)$ such that the series ${\sum }_{n=1}^{\infty }{\epsilon }_n{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon }_n=\pm 1$ and any permutation π of indices.

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we describe the...