Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExpected mean width of the randomized integer convex hull
142
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
rate research
Read More
A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.
Let $x_1,ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $mu $ on ${mathbb R}^n$, and consider the random polytope $$K_N:={rm conv}{ pm x_1,ldots ,pm x_N}.$$ We provide sharp estimates for the quermass{}integrals and other geometric parameters of $K_N$ in the range $cnls Nlsexp (n)$; these complement previous results from cite{DGT1} and cite{DGT} that were given for the range $cnls Nlsexp (sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(mu )$ of $mu $ for all $1ls qls n$.
Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $ell$-norm of convex bodies whose Lowner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschlager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Lowner ellipsoid is the Euclidean unit ball. Here we prove stronger stabili
Let $X_1,ldots,X_n$ be independent random points that are distributed according to a probability measure on $mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,ldots,X_n$ ($ngeq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.
This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
