Œ ⥬ â¨ç­i áâã¤iù
1993, ¢¨¯ã᪠2
“„Š 515.12
ON THE HYPERSPACE OF STRICTLY CONVEX BODIES
L.E. Bazylevych
Abstract. L. Bazylevych, On the hyperspace of strictly convex bodies, Math. Stud.
2 (1993) 83{86.
It is proved that the subset of smooth strictly convex bodies forms a pseudointerior
in the hyperspace of compact convex subsets of the unit cube of Euclidean space.
1. Introduction
"Typical" properties of compact convex subsets of Euclidean spaces are considered in many works (see [1{3]). Here, "typical" means dense Gδ in the hyperspace
of convex compacta. In particular, it is shown in [3] that the subset of smooth and
strictly convex bodies is typical. Recall that a convex body is said to be strictly
convex if its boundary does not contain any segment.
The main result of the given paper contains the description of topology of the
hyperspace of smooth strictly convex bodies. Moreover, it is shown that this hyperspace forms a pseudointerior of the hyperspace of convex compacta which is known
to be homeomorphic to the Hilbert cube Q [4].
2. Preliminaries
For convenience, we deal with convex compacta lying in the unit cube I n =
[−1, 1]n , n ≥ 2. Let M = {A ⊂ I n | A is a convex compactum, A ̸= ∅}. S.B.Nadler,
J.I.Quinn and N.M.Stavrokas [4] have shown that M ∼
= Q = [−1, 1]ω .
Recall that a closed subset A ⊂ Q is called a Z -set [5] if the identity map of Q can
be approximated
by maps into Q\A. A subset A ⊂ Q is called a Z -skeletoid [5] if
∪∞
A = i=1 Ai , where A1 ⊂ A2 ⊂ . . . is a sequence of Z -sets satisfying the condition:
for each ε > 0, n ∈ N and Z -set C ⊂ Q there exists m > n and autohomeomorphism
ψε : Q → Q such that
(1) d(ψε , id) < ε;
(2) ψε |(C ∩ An ) = id;
(3) ψε (C ) ⊂ Am
1991 Mathematics Subject Classification. 54B20, 57N20.
83
Typeset by AMS-TEX
84
L.E. BAZYLEVYCH
(d denoted any xed metric on Q).
We need the following results.
Proposition 1. Let A ⊂ Q be a Z -skeletoid and B ⊃ A be a σ -Z -set (i.e. B is a
countable union of Z -sets) in Q. Then B is a Z -skeletoid.
Theorem 1. Let A ⊂ Q be a Z -skeletoid. Then (Q, Q\A) ∼
= (Q, s), where s =
ω
ω
(−1, 1) is the pseudointerior of the Hilbert cube Q = [−1, 1] .
Proofs. See [5].
3. Main result
Denote by N the subset of M consisting of strictly convex smooth bodies.
Theorem 2. (M, N ) ≡ (Q, s).
Proof. For each k ∈ N let Bk = {A ∈ M | A is a convex body and Bd A contains a
seqment of length ≥ 1/k}, Ck = {A ∈ M | there exists a ∈ Bd A and two tangent
hyperplanes in A with the angle between then ≥ π − 1/k}.
∪∞It is easy to see that Bk and Ck are compact for each k ∈ N. Since M\N =
k=1 (Bk ∪ Ck ), we obtain that N is of the type Gδ in M.
Fix ε > 0 and dene a map Fε : M → N which is 3ε-close to the identity. Let
Oε (A) = {x ∈ Rn | d(x, A) ≤ ε} (here, d is the usual Euclidean metric in Rn ). It is
easy to see that the map Oε is continuous and Oε (A) is (strictly) convex body for
every (strictly) convex A ∈ M.
Let Gε (A) = {(1−ε)x | x ∈ A}. Obviously, the map Gε preserves strict convexity
and smoothness of convex bodies. The composition Oε Gε is a map from M to N
and d(Oε Gε , id) = ε.
Let A ∈ N and a ∈ Bd A. Choose any tangent plane to A in the point a.
Let D be the closure of the half-space of Rn \ containing A. Denote by ba the
point lying in D on the line which passes through the point a orthogonally to and d(a, ba ) = max{2, (ε2 + 4)/2ε} = rε . ∩
Dene the map Hε , Hε (A) = A ∩ {B (ba , rε ) | a ∈ Bd A}, where A is a
compact convex body. It is easy to see that Hε (A) is a stricly convex body. Indeed,
assume that Bd(Hε (A)) contains a segment [m, n]. Obiously, no subsegment of
[m, n] is contained in Bd A. There exists an interval (m1 , n1 ) ⊂ [m, n] such that
(m1 , n1 ) ∩ Bd A = ∅. Let k ∈ (m1 , n1 ). Then there exists a ball B (ba , rε ) containing
k . But then (m1 , n1 ) ⊂ B (ba , ra ) and we get a contradiction.
Obviously, d(Hε (A), A) ≤ ε and immediate calculations show that Hε is a continuous map.
Finally, let Fε = Oε Gε Hε Oε Gε . We have d(Fε , idM ) ≤ 3ε.
We obtain, by arbitrarity of ε, that each compact subset of M\N is a Z -set in
M. Hence, M\N is a σ -Z -set in M.
Now we show that M\N contains a Z -skeletoid. For each e ∈ S n−1 = {x ∈ Rn |
∥x∥ = 1} denote by l(e) the line passing through 0 and e. By pre (A), A ∈ M, we
denote the projection of A onto l(e). Clearly, pre (A) is a segment and we denote it
by [aA (e), bA (e)] (here we assume that bA (e) − aA (e) = λe, where λ > 0). Denote
by e (a), a ∈ l(e), the hyperplane passing through the point a orthogonally to l(e)
and let De (a) be the closure of the halfspace of R\e (a) such that 0 ∈ De (a).
Let e0 = (1, 0, . . . , 0) ∈ S n−1 and An = {A ∈ M | diam(e0 (bA (e0 ) ∩ Bd A) ≥
1/n}.
ON THE HYPERSPACE OF STRICTLY CONVEX BODIES
85
Now we show that A = ∪∞
n=1 An is a Z -skeletoid. Given ε > 0 and n ∈ N,
we construct a homeomorphism ψε satisfying the conditions of the denition of
Z -skeletoid.
Denoting by ρ the Hausdor distance, we have for every A ∈ M the function
ρ(A, An ) = inf {ρ(A, B ) | B ∈ An }.
Dene the map F : M → N ∪ An by the formula:


 Fε/3 (A),
F (A) = F2d(A) (A),


A,
if d ≥ ε/6,
if 0 < d(A) < ε/6,
if A ∈ An ,
where d(A) = ρ(A, An ).
Note that d(A) ≥ ε/6 implies that F (A) contains a ball of radius ε/3. Let
d(A) < ε/6. Then there exists B ∈ An such that ρ(A, B ) = d(A), i.e. Od(A) (A) ⊃
B . Hence, we obtain that for any A ∈ M the set F (A) contains a segment of length
kA = min{1/n, 2ε/3} lying in a hyperplane parallel to e0 (e0 ).
Let φAe : [0, bF (A) (e) − aF (A) (e)] → [0, 2] be a function dened by the condition: φAe (h) = diam(F (A) ∩ e (δeA (h))), where δeA (h) is the point of the line l(e),
d(δeA (h), bF (A) (e)) = h and bF (A) (e) − δeA (h) = λ e for some λ > 0.
If A ∈ M\An , then F (A) ∈ N implies that there exists the maximal number
F (A)
mA
(e) − aF (A) (e)) such that the function φ~Ae = φAe |[0, mA ] strictly
e ∈ (0, b
increases from 0 to nAe , where nAe ≥ min{4ρ(A, An ), 2/3ε} = keA . This implies that
A
keA0 ≥ kA . For xed e, the numbers mA
e , ne continuously depend on A.
Let ψ : M → R be a function dened by the formula:


 1/m,
ψ (A) = (1/m) + 6((1/n) − (1/m))ρ(A, An )/ε,


1/n,
if ρ(A, An ) ≥ ε/6,
if 0 < ρ(A, An ) < ε/6,
if A ∈ An .
Dene the map : M → An by the formula:
(A) = F (A) ∩ De0 (δeA (φ~Ae0 )−1 (ψ(A))).
o
The map satises the conditions (1){(3) in the denition of Z -skeletoid but it
fails to be a homeomorphism.
In order∏to∞ modify the map to homeomorphism, choose a homeomorphism
M∼
= Q = i=1 [0, 2−i ] and denote by (xiA )∞
i=1 the image of A ∈ M under this homeomorphism. For A ∈ M\An denote by ei (A) the point (cos αi (A), sin αi(A), 0, . . . , 0)
∈ S n−1 , where
i
(∑
αi (A) = π
2−j−1 ) + π xiA 2−i+3 .
j =1
)
F (A) ~ F (A) −1
Let ψ(A) = (A) ∩ ∩∞
(2d(A) sin(π/2i+4 )) , where
i=1 De1 (A) (δe (A) )(e (A) )
i
d(A) =
{
ε/3,
2ρ(A, An ),
and let ψ(A) = A for A ∈ An .
i
if ρ(A, An ) ≥ ε/6,
if 0 < ρ(A, An ) < ε/6.
86
L.E. BAZYLEVYCH
It easily follows from the denition of ψ that for each A ∈ M\An the boundary
of ψ(A) contains an innite family of "at" subsets, i.e. intersections
Bd ψ(A) ∩
∏
)
(δeF ((AA)) )(~ Fe ((AA)) )−1 (2d(A) sin(π/2i+4 )) .
i
i
ei (A)
This implies that the sequence (ei (A))∞
i=1 completely determines the point A ∈
M\An . Thus, ψ is required map.
Since M\N contains the Z -skeletoid A = ∪∞
n=1 An , we obtain, by Proposition
1, that M\N is a Z -skeletoid and consequently (M, N ) ∼
= (Q, s).
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Institute of Applied Problems in Mechanics and Mathematics, Naukova 3b, Lviv, 290601,
Ukraine
Received 20.01.1991