Abstract
AbstractFor fixed positive integer n, $$p\in [0,1)$$
p
∈
[
0
,
1
)
, $$a\in (0,1)$$
a
∈
(
0
,
1
)
, we prove that if a function $$g:{\mathbb {S}}^{n-1}\rightarrow {\mathbb {R}}$$
g
:
S
n
-
1
→
R
is sufficiently close to 1, in the $$C^a$$
C
a
sense, then there exists a unique convex body K whose $$L_p$$
L
p
curvature function equals g. This was previously established for $$n=3$$
n
=
3
, $$p=0$$
p
=
0
by Chen et al. (Adv Math 411(A):108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368:107166, 2020). Related, we show that if $$p=0$$
p
=
0
and $$n=4$$
n
=
4
or $$n\le 3$$
n
≤
3
and $$p\in [0,1)$$
p
∈
[
0
,
1
)
, and the $$L_p$$
L
p
curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies $$\lambda ^{-1}\le g\le \lambda $$
λ
-
1
≤
g
≤
λ
, for some $$\lambda >1$$
λ
>
1
, then $$\max _{x\in {\mathbb {S}}^{n-1}}h_K(x)\le C(p,\lambda )$$
max
x
∈
S
n
-
1
h
K
(
x
)
≤
C
(
p
,
λ
)
, for some constant $$C(p,\lambda )>0$$
C
(
p
,
λ
)
>
0
that depends only on p and $$\lambda $$
λ
. This also extends a result from Chen et al. [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the $$L_p$$
L
p
surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the $$L_p$$
L
p
-Minkowksi problem, for $$-n<p<0$$
-
n
<
p
<
0
.
Funder
Hungarian Scientific Research Fund
Publisher
Springer Science and Business Media LLC