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Register a new userSections of convex bodies in Johns and minimal surface area position
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We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in Johns position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in Johns position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.
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Let $K$ be an isotropic symmetric convex body in ${mathbb R}^n$. We show that a subspace $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ with probability greater than $1-exp (-sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(mu )$ of an isotropic log-concave probability measure $mu $ on ${mathbb R}^n$. For every $1leq qleq n$ and $gammain (0,1)$ we show that a random subspace $Fin G_{n,(1-gamma )n}$ satisfies $Z_q(mu )cap Fsubseteq c_2(gamma )sqrt{q},B_2^ncap F$. We also give bounds on the diameter of random projections of $Z_q(mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${mathbb R}^n$ then for a random subspace $F$ of dimension $(log n)^4$ one has that all directions in $F$ are sub-Gaussian with constant $O(log^2n)$.
We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${mathbb R}^n$ with $0in {rm int}(K)$ and if $mu $ is a measure on ${mathbb R}^n$ with a locally integrable non-negative density $g$ on ${mathbb R}^n$, then begin{equation*}mu (K)leq left (csqrt{n-k}right )^kmax_{Fin G_{n,n-k}}mu (Kcap F)cdot |K|^{frac{k}{n}}end{equation*} for every $1leq kleq n-1$. Also, if $mu $ is even and log-concave, and if $K$ is a symmetric convex body in ${mathbb R}^n$ and $D$ is a compact subset of ${mathbb R}^n$ such that $mu (Kcap F)leq mu (Dcap F)$ for all $Fin G_{n,n-k}$, then begin{equation*}mu (K)leq left (ckL_{n-k}right )^{k}mu (D),end{equation*} where $L_s$ is the maximal isotropic constant of a convex body in ${mathbb R}^s$. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals.
The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to $p$-affine surface areas.
Let $d ge 2$, and let $K subset {Bbb{R}}^d$ be a convex body containing the origin $0$ in its interior. In a previous paper we have proved the following. The body $K$ is $0$-symmetric if and only if the following holds. For each $omega in S^{d-1}$, we have that the $(d-1)$-volume of the intersection of $K$ and an arbitrary hyperplane, with normal $omega$, attains its maximum if the hyperplane contains $0$. An analogous theorem, for $1$-dimensional sections and $1$-volumes, has been proved long ago by Hammer (cite{H}). In this paper we deal with the ($(d-2)$-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small $C^2$-perturbations, or $C^3$-perturbations of the Euclidean unit ball, respectively.
We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.
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