对偶平均Minkowski对称度在Minkowski-临界点处的精确值

f
沂C *
a
[0袁1]
s沂Ca[0袁1]
f 渊x*冤 渊in
turn
as肄 渊C冤 =
第 35 卷第 1 期 圆园18 年 3 月
苏 州 科 技 大 学 学 报 渊自 然 科 学 版冤
允燥怎则灶葬造 燥枣 杂怎扎澡燥怎 哉灶蚤增藻则泽蚤贼赠 燥枣 杂糟蚤藻灶糟藻 葬灶凿 栽藻糟澡灶燥造燥早赠 渊晕葬贼怎则葬造 杂糟蚤藻灶糟藻冤
doi院10.12084/j.issn.2096-3289.2018.01.004
critical Minkowski points of a convex body. First袁we established a theorem of Helly for families of half-spaces. Then with
the help of this theorem袁we confirmed the existence of critical n-support configurations. Finally袁in terms of the Minkows鄄
ki measure of asymmetry袁we gave a formula for the precise value of the dual mean Minkowski measure functions at the
critical points of a convex body.
Key words院 Minkowski measure of symmetry曰dual mean Minkowski measure of symmetry曰critical point曰critical support
Abstract院 We investigated the properties of the dual mean Minkowski measure functions of symmetry袁in particular, the
existence of critical n -support configurations and the value of the nth -dual mean Minkowski measure functions at the
and the Minkowski measure of asymmetry as肄渊C冤 of C is so defined as
as肄渊C冤院=infx沂int
C
as肄渊x冤=
1-supx沂int supx沂int C
inf f渊x冤 C
f沂Ca[0袁1]
inf f渊x冤 f沂
Ca [0袁1]
第1期
刘建国袁等院对偶平均 Minkowski 对称度在 Minkowski原临界点处的精确值
17
A point x*沂int C such that as肄渊x*冤=as肄渊C冤 is called a 肄 -critical point 渊or Minkowski-critical point by
Ca[0袁1]院={f | f 院 n寅 is an affine function with f渊C冤=[0袁1]}
One of the important geometric invariants of convex bodies袁the Minkowski measure function of asymmetry as肄(窑)袁
is defined by
as肄渊x冤=as肄渊C袁x冤院=sup
f沂Ca[0袁1]
1-f渊x冤 f渊x冤
=
1-inf
f沂
Ca [0袁1]
f渊x冤
inf f渊x冤 f沂
Ca [0袁1]
袁x沂int
C
where 野int冶 denotes the interior (the definition of this form is from [1]袁see [2-4] for other equivalent definitions)袁
要要要要要要要要要要要要要 要要要要要要
眼收稿日期演 圆园17原04原01 眼基金项目演 国家自然科学基金资助项目渊11271282曰11671293冤 眼作者简介演 刘建国渊1985原冤袁男袁江苏兴化人袁硕士研究生袁研究方向院凸几何分析遥
* 通信作者院国 起渊1957原冤袁男袁博士袁教授袁硕士生导师袁E原mail院guoqi@遥
Let C奂 n渊the n-dimensional Euclidean space冤 be a convex body渊i.e. a compact convex set with non-empty
interior冤.
Denote
by
Ca [0袁1]
the
determining
family
of
C袁i.e.[1]
灾燥造援35 晕燥援1 Mar援 圆园18
The value of dual mean Minksowski measure of symmetry at the critical Minkowski points
of a convex body
LIU Jianguo袁 GUO Qi*
渊School of Mathematics and Physics袁SUST袁Suzhou 215009袁China冤
other authors冤. The set of all 肄-points of C is called the 肄-critical set of C袁denoted by C肄袁which is a nonempty
closed
convex
subset
of
C [2 -3].
Given
x* 沂C肄袁an
configuration
CLC number院 O18曰O174.13
酝砸渊圆园1园冤 杂怎遭躁藻糟贼 悦造葬泽泽蚤枣蚤糟葬贼蚤燥灶院 52A01曰52A30曰52A38
Document code院 粤
Article ID院 2096原3289渊圆园18冤园1原园园16原园5
1 Introduction