Bezout Inequality for Mixed Volumes
نویسندگان
چکیده
In this paper we consider the following analog of Bezout inequality for mixed volumes: V (P1, . . . , Pr,∆ )Vn(∆) r−1 ≤ r ∏ i=1 V (Pi,∆ ) for 2 ≤ r ≤ n. We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional simplex. We prove that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to ∆), which confirms the conjecture when ∆ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.
منابع مشابه
Definability and Fast Quantifier Elimination in Algebraically Closed Fields
The Bezout-Inequality, an afine version (not in&ding multiplicities) of the classical Bezout-Theorem is derived for applications in algebraic complexity theory. Upper hounds for the cardinality and number of sets definable by first order formulas over algebraically closed fields are given. This is used for fast quantifier elimination in algebraically closed fields.
متن کاملDesign of FIR Precoders and Equalizers for Broadband MIMO Wireless Channels with Power Constraints
This paper examines the optimum design of FIR precoders or equalizers for multiple-input multiple-output (MIMO) frequencyselective wireless channels. For the case of a left-coprime FIR channel, which arises generically when the number nT of transmit antennas is larger than the number nR of receive antennas, the Bezout matrix identity can be employed to design an FIR MIMO precoder that equalizes...
متن کاملLp-Minkowski and Aleksandrov-Fenchel type inequalities
In this paper we establish the Lp-Minkowski inequality and Lp-Aleksandrov-Fenchel type inequality for Lp-dual mixed volumes of star duality of mixed intersection bodies, respectively. As applications, we get some related results. The paper new contributions that illustrate this duality of projection and intersection bodies will be presented. M.S.C. 2000: 52A40.
متن کاملVolume difference inequalities for the projection and intersection bodies
In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.
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 2015