Article
Kyungpook Mathematical Journal 2020; 60(3): 485-505
Published online September 30, 2020
Copyright © Kyungpook Mathematical Journal.
Extreme Points, Exposed Points and Smooth Points of the Space
Sung Guen Kim
Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : sgk317@knu.ac.kr
Received: September 5, 2019; Revised: March 10, 2020; Accepted: April 21, 2020
Abstract
We present a complete description of all the extreme points of the unit ball of
Keywords: symmetric bilinear forms on ℝ,3 with the supremum norm, extreme points, exposed points, smooth points.
1. Introduction
We denote by
In 1998, Choi
In 2009, Kim [12] initiated the classification of the extreme points, exposed points, and smooth points of the unit ball of
where
Let
If
as a 4 × 1 column vector. If
as a 3 × 1 column vector. If
as a 9 × 1 column vector. If
as a 6 × 1 column vector.
In [12] it was shown:
-
(a)
; -
(b)
.
We refer to ([1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] for some recent works about extremal properties of multilinear mappings and homogeneous polynomials on some classical Banach spaces.
In this paper we present a complete description of the 42 extreme points of the unit ball of
2. The Extreme Points of the Unit Ball of
Recently, Kim [22] showed the following: Let
and
The following statements hold true.
-
(a) Let
with || T || = 1. Then,if and only if there exist at least 6 linearly independent vectors W 1, …,W 6 ∈ Ω andZ 1, …,Z 6 ∈ Γ such thatLet
A be the invertible 6 × 6 matrix such that thej -row vector ofA as [Row (A )]j =W j forj = 1, …, 6. Then,AS = (S (Z 1), · · ·,S (Z 6))t for all. -
(b)
.
Theorem 2.1.([22])
Let
Note that if ||
Theorem 2.2
Claim:
and
with ||
which show that
Claim:
and
with ||
which show that
The other 40 bilinear forms in the list of Theorem 2.2 can be proved to be extreme in a similar way. We leave this to the reader.
Let
for some
Those
Theorem 2.3
Let
Suppose that
Write
and
for some
Note that, for
and
Since
and
It follows that
which imply that, for all |
which shows that
Suppose that for
and
for some
which shows that, for |
If
which shows that
We claim that
Let
Suppose that
Write
and
for some
and
Since
and
It follows that
which imply that, for all |
which shows that
Suppose that for
and
for some
which shows that, for |
By Claim 1, we conclude that
Let
Suppose that
Write
and
for some
which imply that
and
for every |
Notice that the other 39 cases are similar since, essentially there are three groups of extreme points, those having an 1, those having two
3. The Exposed Points of the Unit Ball of
Theorem 2.2 leads to a complete formula of ||
Theorem 3.1
By the Krein-Milman Theorem,
Note that if ||
Theorem 3.2.([17])
We give another proof of the following theorem which was shown in [22].
Theorem 3.3.([22])
It suffices to show that if
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
Claim:
Let
4. The Smooth Points of the Unit Ball of
In this section we will characterize all the smooth points of the unit ball of
Theorem 4.1
(⇐):
By Theorem 2.1, ||
Without loss of generality, we may assume that
Let
We claim that
By Theorem 2.1, for
It follows that for
-
(1)
-
(2)
-
(3)
-
(4)
By (1) − (4),
By Theorem 2.1, for
Since
so,
hence,
By Theorem 2.1, ||
Let
Let
and
By Theorem 2.1, for
It follows that for
-
(1′)
-
(2′)
-
(3′)
-
(4′)
-
(5′)
By (1′) − (5′),
hence,
(⇒): If not, then we have two cases.
for
Notice that
We have ten subcases as follows:
Suppose that
and
By Theorem 3.1, ||
Suppose that
By Theorem 3.1, ||
Suppose that
By Theorem 3.1, ||
Suppose that
and
By Theorem 3.1, ||
Suppose that
and
By Theorem 3.1, ||
Suppose that
and
By Theorem 3.1, ||
Since the proofs of other subcases are similar as those of the above, we omit the proofs.
There exist
for
Acknowledgements
The author is thankful to the referee for the careful reading and considered suggestions leading to a better presented paper.
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