Nous étudions des analogues en dimension supérieure de l’inégalité de Burago $A\left(S\right)\le R^{2}T\left(S\right)$, avec $S$ une surface fermée de classe $C^2$ immergée dans ${ℝ}^3$, $A\left(S\right)$ son aire et $T\left(S\right)$ sa courbure totale. Nous donnons un exemple explicite qui prouve qu’une inégalité analogue de la forme $\mathrm{vol}\left(M\right)\le C_n{R}^{n}T\left(M\right)$, avec $C_n\>0$ une constante, ne peut être vraie pour une hypersurface fermée $M$ de classe $C^2$ dans ${ℝ}^{n+1}$, $n\ge 3$. Nous mettons toutefois en évidence une condition suffisante sur la courbure de Ricci sous laquelle l’inégalité est vérifiée en dimension $n=3$. En dimension...

There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $in{f}_{ellipsoid\epsilon \subset B_E}{(vol\left(B_E\right)/vol\left(\epsilon \right))}^{1/n}\le c_{0}in{f}_{zonoidZ\subset B_F}{(vol\left(B_F\right)/vol\left(Z\right))}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_p$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_p,dimF=n}in{f}_{G\subset L_q,dimG=n}d(F,G){\sim }_{c_{pq}}{n}^{(q/2)(1/q-1/p)}$.

Let g be a Gaussian random vector in ℝⁿ. Let N = N(n) be a positive integer and let $K_N$ be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes $V_N:=vol\left(K_N\cap RB₂ⁿ\right)/vol\left(RB₂ⁿ\right)$. For a large range of R = R(n), we establish a sharp threshold for N, above which $V_N\to 1$ as n → ∞, and below which $V_N\to 0$ as n → ∞. We also consider the case when $K_N$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R ∈ (0,1) and...

The geometry of random projections of centrally symmetric convex bodies in ${ℝ}^N$ is studied. It is shown that if for such a body K the Euclidean ball $B{₂}^N$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B=P_H\left(K\right)$ is “far” from $P_H\left(B{₂}^N\right)$ then the (random) body B is as “rigid” as its “distance” to $P_H\left(B{₂}^N\right)$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).