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Miklós Schweitzer
2019 Miklós Schweitzer
5
5
Part of
2019 Miklós Schweitzer
Problems
(1)
Inequality related to defining convex body from halfspaces
Source: Miklós Schweitzer 2019, Problem 5
12/27/2019
Let
S
⊂
R
d
S \subset \mathbb{R}^d
S
⊂
R
d
be a convex compact body with nonempty interior. Show that there is an
α
>
0
\alpha > 0
α
>
0
such that if
S
=
∩
i
∈
I
H
i
S = \cap_{i \in I} H_i
S
=
∩
i
∈
I
H
i
, where
I
I
I
is an index set and
(
H
i
)
i
∈
I
(H_i)_{i \in I}
(
H
i
)
i
∈
I
are halfspaces, then for any
P
∈
R
d
P \in \mathbb{R}^d
P
∈
R
d
, there is an
i
∈
I
i \in I
i
∈
I
for which
d
i
s
t
(
P
,
H
i
)
≥
α
d
i
s
t
(
P
,
S
)
\mathrm{dist}(P, H_i) \ge \alpha \, \mathrm{dist}(P, S)
dist
(
P
,
H
i
)
≥
α
dist
(
P
,
S
)
.
inequalities