DOI: 10.68381/jca14047 ISSN: 0944-6532
Hausdorff Dimension of Cut Loci of Generic Subspaces of Euclidean Spaces
Alain Rivière
Let
F
F
be a closed set of the Euclidean space
\Bbb{E}^d
E
d
, with
\emptyset\not=F\not=\Bbb{E}^d
∅
≠
F
≠
E
d
and
d\geq 2
d
≥
2
. Let
\mathcal{N}
N
be the set of centers of all open balls contained in
\Bbb{E}^d \setminus F
E
d
∖
F
which are maximal with respect to inclusion. We prove that the Hausdorff dimension
\mathrm{dim_H}(\mathcal{N})
d
i
m
H
(
N
)
of
\mathcal{N}
N
equals
d
d
when
F
F
is, in the sense of Baire categories, a generic compact subset of
\mathbb{E}^d
E
d
, or when
\Bbb{E}^d \setminus F
E
d
∖
F
is the interior of a generic convex body of
\mathbb{E}^d
E
d
. If
C
C
is a generic convex body, we deduce that the set of all points of
\partial C
∂
C
where the “upper curvature” of
\partial C
∂
C
is positive and finite, is of Hausdorff dimension
d-1
d
−
1
. Let
\mathrm{CurvCt}
C
u
r
v
C
t
be the set of centers of upper curvature of
\partial C
∂
C
, and
\omega
ω
be any non empty open subset of
\Bbb{E}^d
E
d
. We also prove that
\mathop{{\rm dim}_{\rm H}}\nolimits(\omega\cap \mathrm{CurvCt})=d
d
i
m
H
(
ω
∩
C
u
r
v
C
t
)
=
d
. Let
B
B
be a generic compact subset of
\mathbb{E}^d
E
d
, or a generic convex body of
\mathbb{E}^d
E
d
. Let
\mathrm{a}\mathcal{N}
a
N
be the set of centers of all closed balls containing
B
B
which are minimal with respect to inclusion. We also prove that
\mathrm{dim_H}(\mathrm{a}\mathcal{N})=d
d
i
m
H
(
a
N
)
=
d
. The proofs employ some of the ideas used in a previous paper of the author [“Dimension de Hausdorff de la nervure”, Geom. Dedicata, 85 (2001) 217–235] to construct large cut loci in
\mathbb{E}^d
E
d
.