اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn extremums of sums of powered distances to a finite set of points
269
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we investigate the extremal properties of the sum $$sum_{i=1}^n|MA_i|^{lambda},$$ where $A_i$ are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and $M$ varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on $Gamma$ the extremal values of the sum are obtained in terms of $lambda$. In the case of the regular dodecahedron and icosahedron in $mathbb{R}^3$ we obtain results for which values of $lambda$ the corresponding sum is independent of the position of $M$ on $Gamma$. We use elementary analytic and purely geometric methods.
قيم البحث
اقرأ أيضاً
In this paper we consider an extremal problem in geometry. Let $lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $Gamma$. We give full characterization of the extremal behavior of the function $f(M,lambda)=MA^lambd
a+MB^lambda+MC^lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric methods in the proof.
We consider the number of distinct distances between two finite sets of points in ${bf R}^k$, for any constant dimension $kge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such
that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Omegaleft(minleft{ n^{2/3}m^{2/3},; frac{n^{10/11}m^{4/11}}{log^{2/11}m},; n^2,; m^2right}right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.
Our starting point is an improved version of a result of D. Hajela related to a question of Koml{o}s: we show that if $f(n)$ is a function such that $limlimits_{ntoinfty }f(n)=infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every $ng
eqslant n_0$ and any $Ssubseteq {-1,1}^n$ with cardinality $|S|leqslant 2^{n/f(n)}$ one can find orthonormal vectors $x_1,ldots ,x_nin {mathbb R}^n$ that satisfy $$|epsilon_1x_1+cdots +epsilon_nx_n|_{infty }geqslant csqrt{log f(n)}$$ for all $(epsilon_1,ldots ,epsilon_n)in S$. We obtain analogous results in the case where $x_1,ldots ,x_n$ are independent random points uniformly distributed in the Euclidean unit ball $B_2^n$ or any symmetric convex body, and the $ell_{infty }^n$-norm is replaced by an arbitrary norm on ${mathbb R}^n$.
The problem of finding near-stationary points in convex optimization has not been adequately studied yet, unlike other optimality measures such as minimizing function value. Even in the deterministic case, the optimal method (OGM-G, due to Kim and Fe
ssler (2021)) has just been discovered recently. In this work, we conduct a systematic study of the algorithmic techniques in finding near-stationary points of convex finite-sums. Our main contributions are several algorithmic discoveries: (1) we discover a memory-saving variant of OGM-G based on the performance estimation problem approach (Drori and Teboulle, 2014); (2) we design a new accelerated SVRG variant that can simultaneously achieve fast rates for both minimizing gradient norm and function value; (3) we propose an adaptively regularized accelerated SVRG variant, which does not require the knowledge of some unknown initial constants and achieves near-optimal complexities. We put an emphasis on the simplicity and practicality of the new schemes, which could facilitate future developments.
Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${mathbb R}^n$. We provide upper bounds for the multi-integral expression begin{equation*}|{bf t}|_{C^s,K}=int_{C}cdotsint_{C}Big|sum_{j=1}^st_jx_jBig|_K,dx_1cdots dx_send{equation
*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to randomized vector balancing problems.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
