Modified Shephard’s Problem on Projections of Convex Bodies
نویسنده
چکیده
We disprove a conjecture of A. Koldobsky asking whether it is enough to compare (n−2)-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions.
منابع مشابه
Projections of Convex Bodies and the Fourier Transform
Abstract. The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the BusemannPetty problem, characterizations of intersection bodies, extremal sections of lp-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the resu...
متن کاملModified Convex Data Clustering Algorithm Based on Alternating Direction Method of Multipliers
Knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering in which there is no need to be peculiar about how to select initial values. Due to properly converting the task of optimization to an equivalent...
متن کاملThe Fourier Transform and Firey Projections of Convex Bodies
In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, that he called p-sums. Lutwak [Lu2], [Lu3] showed that these Firey sums lead to a Brunn-Minkowski theory for each p ≥ 1. He introduced the notions of p-mixed volume, p-surface area measure, and proved an integral representation and inequalities for p-mixed volumes, ...
متن کاملFourier Transform and Firey Projections of Convex Bodies
In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, what he called p-sums. E. Lutwak [Lu2], [Lu3] showed that these Firey sums lead to a Brunn-Minkowski theory for each p ≥ 1. He introduced the notions of p-mixed volume, p-surface area measure, and proved an integral representation and inequalities for p−mixed volume...
متن کاملValuations and Busemann–Petty Type Problems
Projection and intersection bodies define continuous and GL(n) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann– Petty problem. We consider the question whether ΦK ⊆ ΦL implies V (K) ≤ V (L), where Φ is a homogeneous, continuous operator on convex or star bodies which is...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007