On Symmetric Versions of Sylvester’s Problem
نویسنده
چکیده
We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.
منابع مشابه
Single Facility Goal Location Problems with Symmetric and Asymmetric Penalty Functions
Location theory is an interstice field of optimization and operations research. In the classic location models, the goal is finding the location of one or more facilities such that some criteria such as transportation cost, the sum of distances passed by clients, total service time, and cost of servicing are minimized. The goal Weber location problem is a special case of location mode...
متن کاملSome results on the symmetric doubly stochastic inverse eigenvalue problem
The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$, to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$. If there exists an $ntimes n$ symmetric doubly stochastic ...
متن کاملBOUNDS ON KRONECKER AND q-BINOMIAL COEFFICIENTS
We present a lower bound on the Kronecker coefficients of the symmetric group via the characters of Sn, which we apply to obtain various explicit estimates. Notably, we extend Sylvester’s unimodality of q-binomial coefficients ( n k ) q as polynomials in q to derive sharp bounds on the differences of their consecutive coefficients.
متن کاملNon-commutative Sylvester’s determinantal identity, preprint
Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a β-extension that is both a generalization of Sylvester's identity and the β-extension of the MacMahon master theorem.
متن کاملTwo Proofs for Sylvester’s Problem Using an Allowable Sequence of Permutations
The famous Sylvester’s problem is: Given finitely many noncollinear points in the plane, do they always span a line that contains precisely two of the points? The answer is yes, as was first shown by Gallai in 1944. Since then, many other proofs and generalizations of the problem appeared. We present two new proofs of Gallai’s result, using the powerful method of allowable sequences.
متن کامل