We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies somewhat stronger weight
We present an alternative, short proof of a recent discrete version of the Brunn-Minkowski inequality due to Lehec and the second named author. Our proof also yields the four functions theorem of Ahlswede and Daykin as well as some new variants.
A central problem in discrete geometry, known as Hadwigers covering problem, asks what the smallest natural number $Nleft(nright)$ is such that every convex body in ${mathbb R}^{n}$ can be covered by a union of the interiors of at most $Nleft(nright)
$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n choose n}nln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n choose n}e^{-csqrt{n}}$ for some universal constant $c>0$. Our approach combines ideas from previous work by Artstein-Avidan and the second named author with tools from Asymptotic Geometric Analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$; in fact, we get the same lower bound for the volume of the intersection of $K$ and $-K$ when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for $Nleft(nright)$ when restricting our attention to convex bodies that are $psi_{2}$. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity.
We study a new construction of bodies from a given convex body in $mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We sho
w that these bodies are related to Ulams long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.
Given a Borel measure $mu$ on ${mathbb R}^{n}$, we define a convex set by [ M({mu})=bigcup_{substack{0le fle1, int_{{mathbb R}^{n}}f,{rm d}{mu}=1 } }left{ int_{{mathbb R}^{n}}yfleft(yright),{rm d}{mu}left(yright)right} , ] where the union is taken ov
er all $mu$-measurable functions $f:{mathbb R}^{n}toleft[0,1right]$ with $int_{{mathbb R}^{n}}f,{rm d}{mu}=1$. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures.
We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality of separation and covering. We provide analogues for various geometric inequalities on cover
ing numbers, such as volume bounds, bounds connected with Hadwigers conjecture, and inequalities about M-positions for geometric log-concave functions. In particular, we obtain stro
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove several genera
lizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n+2 fixed projective points in real n-dimensional projective space , through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical
and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm Nasz{o}dis equivalent fractional illumination conjecture in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).
