$L_p$ -dual mixed affine surface areas
Lutwak proposed the notion of $L_p$-affine surface area according to the Lp-mixed volume. Recently,Wang and He introduced the concept of Lp-dual affine surface area combing with the $L_p$-dual mixed volume. In the article, we give the concept of $L_p$-dual mixed affine surface areas associated with...
Gespeichert in:
| Datum: | 2016 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1864 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507741491036160 |
|---|---|
| author | Wan, Xiaoyan Wang, W. Ван, Сяоянь Ванг, В. |
| author_facet | Wan, Xiaoyan Wang, W. Ван, Сяоянь Ванг, В. |
| author_sort | Wan, Xiaoyan |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:30:15Z |
| description | Lutwak proposed the notion of $L_p$-affine surface area according to the Lp-mixed volume. Recently,Wang and He introduced the concept of Lp-dual affine surface area combing with the $L_p$-dual mixed volume. In the article, we give the concept of $L_p$-dual mixed affine surface areas associated with the $L_p$-dual mixed quermassintegrals. Further, some inequalities for the $L_p$-dual mixed affine surface areas are obtained. |
| first_indexed | 2026-03-24T02:14:08Z |
| format | Article |
| fulltext |
UDC 517.5
X. Wan, W. Wang (China Three Gorges Univ., Yichang, China)
\bfitL \bfitp -DUAL MIXED AFFINE SURFACE AREAS*
\bfitL \bfitp -DUAL MIXED AFFINE SURFACE AREAS
Lutwak proposed the notion of Lp-affine surface area according to the Lp-mixed volume. Recently, Wang and He introduced
the concept of Lp-dual affine surface area combing (combined ??? PM???) with the Lp-dual mixed volume. In the article,
we give the concept of Lp-dual mixed affine surface areas associated with the Lp-dual mixed quermassintegrals. Further,
some inequalities for the Lp-dual mixed affine surface areas are obtained.
Лутвок запропонував поняття Lp-афiнноЇ поверхневої площi, що вiдповiдає поняттю Lp-змiшаного об’єму. Не-
щодавно Ванг i Хе ввели поняття Lp-дуальної афiнної поверхневої площi, пов’язаної з Lp-дуальним змiшаним
об’ємом. В роботi запропоновано поняття Lp-дуальної змiшаної афiнної поверхневої площi, що вiдповiдає Lp-
дуальним змiшаним квермасiнтегралам. Крiм того, наведено деякi нерiвностi для Lp-дуальних змiшаних афiнних
поверхневих площ.
1. Introduction and main results. Let \scrK n denote the set of convex bodies (compact, convex subsets
with nonempty interiors) in Euclidean space \BbbR n. For the set of convex bodies containing the origin
in their interiors, the set of centroid of convex bodies is the origin and the set of origin-symmetric
convex bodies in \BbbR n, we write \scrK n
o , \scrK n
c and \scrK n
os, respectively. Let \scrS n
o denotes the set star bodies
(about the origin) in \BbbR n. Let Sn - 1 denotes the unit sphere in \BbbR n, denote by V (K) the n-dimensional
volume of body K, for the standard unit ball B in \BbbR n, denote \omega n = V (B).
The studies of the classical affine surface area went back to Blaschke [1]. The notion of classical
affine surface area was extended to convex bodies by Leichtweiß [5]. For K \in \scrK n, the affine surface
area, \Omega (K), of K is defined by
n - 1/n\Omega (K)
n+1
n = \mathrm{i}\mathrm{n}\mathrm{f}\{ nV1(K,Q\ast )V (Q)1/n : Q \in Sn
o \} . (1.1)
Here Q\ast denotes the polar of body Q. Subsequently, Lutwak [10] introduced mixed affine surface
areas. On the researches of classical affine surface areas, also see [6].
The Lp-affine surface areas were introduced by Lutwak [13]: for K \in \scrK n
o , p \geq 1, the Lp-affine
surface area, \Omega p(K), of K is defined by
n - p/n\Omega p(K)
n+p
n = \mathrm{i}\mathrm{n}\mathrm{f}\{ nVp(K,Q\ast )V (Q)p/n : Q \in Sn
o \} .
Here Vp(M,N) denotes the Lp-mixed volume of M,N \in \scrK n
o (see [12, 13]). Obviously, if p = 1,
\Omega p(K) is just the affine surface area \Omega (K) of K.
In addition, Lutwak [13] also gave the notion of Lp-mixed affine surface areas. Moreover, Wang
and Leng in [16] defined Lp-mixed affine surface area, \Omega p,i(K), of K (for i = 0, \Omega p,i(K) is just
the Lp-affine surface area \Omega p(K)) and extended some Lutwak’s results. Regarding the studies of
Lp-affine surface areas, besides see [13, 16], also see [17 – 21]. Recently, Ludwig [7, 8] extended
Lp-affine surface areas to L\phi -affine surface areas.
Because the definition of Lp-affine surface area base on the Lp-mixed volume. In 2008, Wang
and He [14] showed the notion of Lp-dual affine surface area associated with the Lp-dual mixed
* Research is supported in part by the Natural Science Foundation of China (Grant No.11371224).
c\bigcirc X. WAN, W. WANG, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5 601
602 X. WAN, W. WANG
volume. For K \in \scrS n
o , and 1 \leq p < n, the Lp-dual affine surface area, \widetilde \Omega - p(K), of K is defined by
np/n\widetilde \Omega - p(K)
n - p
n = \mathrm{i}\mathrm{n}\mathrm{f}\{ n\widetilde V - p(K,Q\ast )V (Q) - p/n : Q \in \scrK n
c \} . (1.2)
Here \widetilde V - p(M,N) denotes the Lp-dual mixed volume of M,N \in \scrS n
o [13].
Associated with the Lp-dual affine surface areas, Wang and He [14] proved the following dual
forms of Lutwak’s results:
Theorem 1.A. If K \in \scrS n
o , n > p \geq 1, then
\widetilde \Omega - p(K)n - p \geq nn - p\omega - 2p
n V (K)n+p
with equality if and only if K is an ellipsoid.
Theorem 1.B. If K \in \scrK n
os, n > p \geq 1, then
\widetilde \Omega - p(K)\widetilde \Omega - p(K
\ast ) \leq n2\omega 2
n
with equality if and only if K is an ellipsoid.
Theorem 1.C. If K \in \scrS n
o , 1 \leq p \leq q \leq n, then\biggl(
\Omega - p(K)n - p
nn - pV (K)n+p
\biggr) 1/p
\leq
\biggl(
\Omega - q(K)n - q
nn - qV (K)n+q
\biggr) 1/q
.
Here \biggl(
\Omega - p(K)n - p
nn - pV (K)n+p
\biggr) 1/p
(1.3)
be called the Lp-dual affine area ratio of K \in \scrS n
o (see [14]).
Recall that Wang and Leng in [15] extended the notion of Lp-dual mixed volume and gave the
definition of Lp-dual mixed quermassintegrals. The main aim of this article is to define the Lp-dual
mixed affine surface area by the Lp-dual mixed quermassintegrals. Further, we extend Wang and
He’s results.
Now we give the concept of Lp-dual mixed affine surface areas as follows: For K \in \scrS n
o , p \geq 1,
real i \not = n, the Lp-dual mixed affine surface area, \widetilde \Omega - p,i(K), of K is defined by
n
p
n - i \widetilde \Omega - p,i(K)
n - p - i
n - i = \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
n\widetilde W - p,i(K,Q\ast )\widetilde Wi(Q) -
p
n - i : Q \in \scrK n
c
\Bigr\}
. (1.4)
Here \widetilde W - p,i(M,N) denote the Lp-dual mixed quermassintegrals of M,N \in \scrS n
o .
According to definitions (1.2), (1.4) and equality (2.11), we easily know that for K \in \scrK n
os,\widetilde \Omega - p,0(K) = \widetilde \Omega - p(K). (1.5)
Associated with the Lp-dual mixed affine surface areas, we give the general forms of Theo-
rems 1.A, 1.B and 1.C. Our main results can be stated as follows, respectively.
Theorem 1.1. If K \in \scrS n
o , p \geq 1 and 0 \leq i < n, then
\widetilde \Omega - p,i(K)n - p - i \geq nn - p - i\omega - 2p
n
\widetilde Wi(K)n+p - i (1.6)
with equality for i = 0 if and only if K is an ellipsoid, for 0 < i < n if and only if K is a ball.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
Lp-DUAL MIXED AFFINE SURFACE AREAS 603
Theorem 1.2. If K \in \scrK n
c , p \geq 1, and 0 \leq i < n - p, then
\widetilde \Omega - p,i(K)\widetilde \Omega - p,i(K
\ast ) \leq n2\omega 2
n (1.7)
with equality for i = 0 if and only if K is an ellipsoid, for 0 < i < n if and only if K is a ball.
Theorem 1.3. If K \in \scrS n
o , 1 \leq p \leq q, i is a real and 0 \leq i < n, then\Biggl( \widetilde \Omega - p,i(K)n - p - i
nn - p - i\widetilde Wi(K)n+p - i
\Biggr) 1/p
\leq
\Biggl( \widetilde \Omega - q,i(K)n - q - i
nn - q - i\widetilde Wi(K)n+q - i
\Biggr) 1/q
. (1.8)
Similar to the definitions (1.1) and (1.3),\Biggl( \widetilde \Omega - p,i(K)n - p - i
nn - p - i\widetilde Wi(K)n+p - i
\Biggr) 1/p
may be called the Lp-dual mixed affine area ratio of K \in \scrS n
o .
Finally, we give the following Brunn – Minkowski-type inequality for the Lp-dual mixed affine
surface areas.
Theorem 1.4. If K,L \in Sn
o , p \geq 1 and \lambda , \mu \geq 0 (not both zero), real i < n+2p and i \not = n, n+p,
then \widetilde \Omega - p,i(\lambda \cdot K + - p \mu \cdot L) -
p
n+p - i \geq \lambda \widetilde \Omega - p,i(K)
- p
n+p - i + \mu \widetilde \Omega - p,i(L)
- p
n+p - i (1.9)
with equality if and only if K and L are dilates. Here \lambda \cdot K + - p \mu \cdot L denotes the Lp-harmonic
radial combination of K and L.
2. Preliminaries.
2.1. Radial function and polar of convex bodies. If K is a compact star-shaped (about the
origin) in Rn, then its radial function, \rho K = \rho (K, \cdot ) : Rn \setminus \{ 0\} - \rightarrow [0,\infty ), is defined by (see [2, 20])
\rho (K,u) = \mathrm{m}\mathrm{a}\mathrm{x}\{ \lambda \geq 0 : \lambda \cdot u \in K\} , u \in Sn - 1.
If \rho K is continuous and positive, then K will be called a star body. Two star bodies K,L are said to
be dilates (of one another) if \rho K(u)/\rho L(u) is independent of u \in Sn - 1.
If K \in \scrK n
o , the polar body, K\ast , of K is defined by (see [2, 20])
K\ast = \{ x \in Rn : x \cdot y \leq 1, y \in K\} .
Obviously, for K \in \scrK n
o ,
(K\ast )\ast = K. (2.1)
2.2. \bfitL \bfitp -dual mixed quermassintegrals. For K,L \in Sn
o , p \geq 1 and \lambda , \mu \geq 0 (not both zero),
the Lp-harmonic radial combination, \lambda \cdot K + - p \mu \cdot L \in Sn
o , of K and L is defined by (see [13])
\rho (\lambda \cdot K + - p \mu \cdot L, \cdot ) - p = \lambda \rho (K, \cdot ) - p + \mu \rho (L, \cdot ) - p. (2.2)
For K \in \scrS n
o and any real i, the dual quermassintegrals, \widetilde Wi(K), of K are defined by (see [9])
\widetilde Wi(K) =
1
n
\int
Sn - 1
\rho (K,u)n - idS(u). (2.3)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
604 X. WAN, W. WANG
Obviously, \widetilde W0(K) =
1
n
\int
Sn - 1
\rho (K,u)ndS(u) = V (K). (2.4)
For K \in \scrK n
o and its polar body, Ghandehari (see [3]) established an upper bound of the dual
quermassintegrals product as follows:
Theorem 2.A. If K \in \scrK n
c , i is any real and 0 \leq i < n, then
\widetilde Wi(K)\widetilde Wi(K
\ast ) \leq \omega 2
n (2.5)
with equality for i = 0 if and only if K is an ellipsoid centered at the origin, for 0 < i < n if and
only if K is a ball centered at the origin.
Note that the case i = 0 of (2.5) is just the well-known Blaschke – Santaló inequality (see [4]).
Associated with the Lp-harmonic radial combination of star bodies, Wang and Leng (see [15])
introduced the notion of Lp-dual mixed quermassintegrals as follows: for K,L \in Sn
o , p \geq 1, \varepsilon > 0,
real i \not = n, the Lp-dual mixed quermassintegrals, \widetilde W - p,i(K,L), of the K and L be defined by
n - i
- p
\widetilde W - p,i(K,L) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon - \rightarrow 0+
\widetilde Wi(K+ - p\varepsilon \cdot L) - \widetilde Wi(K)
\varepsilon
. (2.6)
The definition above and Hospital’s role give the following integral representation of the Lp-dual
mixed quermassintegrals (see [15]):
\widetilde W - p,i(K,L) =
1
n
\int
Sn - 1
\rho n+p - i
K (u)\rho - p
L (u)dS(u), (2.7)
where the integration with respect to spherical Lebesgue measure S on Sn - 1. From the formula (2.7)
and definition (2.3), we get \widetilde W - p,i(K,K) = \widetilde Wi(K). (2.8)
Theorem 2.B. Let K,L \in \scrS n
o , p \geq 1, and real i \not = n, then for i < n or n < i < n+ p
\widetilde W - p,i(K,L) \geq \widetilde Wi(K)
n+p - i
n - i \widetilde Wi(L)
- p
n - i , (2.9)
for i > n + p inequality (2.9) is reverse. Equality holds in every inequality if and only if K and L
are dilates. For i = n+ p, (2.9) is identic.
Recall that Lutwak in [13] gave the concept of Lp-dual mixed volume: For K,L \in Sn
o , p \geq 1,
the Lp-dual mixed volume, \widetilde V - p(K,L), of the K and L is defined by
n
- p
\widetilde V - p(K,L) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon - \rightarrow 0+
V (K + - p \varepsilon \cdot L) - V (K)
\varepsilon
. (2.10)
From (2.10), (2.6) and (2.4), we see that
\widetilde W - p,0(K,L) = \widetilde V - p(K,L). (2.11)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
Lp-DUAL MIXED AFFINE SURFACE AREAS 605
3. \bfitL \bfitp -Dual Mixed Affine Surface Areas. In this section, we will complete the proofs of
theorems.
Proof of Theorem 1.1. For i = 0, Theorem 1.1 is just Theorem 1.A.
For 0 < i < n, from (2.9) and (2.5), we have
\widetilde W - p,i(K,Q\ast )\widetilde Wi(Q) -
p
n - i \geq \widetilde Wi(K)
n+p - i
n - i
\Bigl[ \widetilde Wi(Q
\ast )\widetilde Wi(Q)
\Bigr] - p
n - i \geq
\geq \omega
- 2p
n - i
n
\widetilde Wi(K)
n+p - i
n - i .
Hence, using definition (1.4), we know
\widetilde \Omega - p,i(K)
n - p - i
n - i \geq n
n - p - i
n - i \omega
- 2p
n - i
n
\widetilde Wi(K)
n+p - i
n - i ,
this yield inequality (1.6).
According to the equality condition of (2.5), we see that equality hold in (1.6) if and only if K
is a ball when 0 < i < n.
Theorem 1.1 is proved.
Proof of Theorem 1.2. For the case i = 0, the proof of Theorem 1.2 see Theorem 1.B.
For 0 < i < n - p, from definition (1.4), it follows that for Q \in \scrK n
c ,
\widetilde \Omega - p,i(K)
n - p - i
n - i \leq n
n - p - i
n - i \widetilde W - p,i(K,Q\ast )\widetilde Wi(Q) -
p
n - i .
Since K \in \scrK n
c , taking K\ast for Q and using (2.1), we can get
\widetilde \Omega - p,i(K)n - p - i \leq nn - p - i\widetilde W - p,i(K,K)n - i\widetilde Wi(K
\ast ) - p.
Thus by (2.8), \widetilde \Omega - p,i(K)n - p - i \leq nn - p - i\widetilde Wi(K)n - i\widetilde Wi(K
\ast ) - p. (3.1)
Similarly, \widetilde \Omega - p,i(K
\ast )n - p - i \leq nn - p - i\widetilde Wi(K
\ast )n - i\widetilde Wi(K) - p. (3.2)
From (3.1) and (3.2), we obtain\Bigl[ \widetilde \Omega - p,i(K)\widetilde \Omega - p,i(K
\ast )
\Bigr] n - p - i
\leq n2(n - p - i)
\Bigl[ \widetilde Wi(K)\widetilde Wi(K
\ast )
\Bigr] n - p - i
.
Hence, using (2.5), we have \Bigl[ \widetilde \Omega - p,i(K)\widetilde \Omega - p,i(K
\ast )
\Bigr] n - p - i
\leq
\leq n2(n - p - i)
\biggl[
\omega
2i
n
n (V (K)V (K\ast ))
n - i
n
\biggr] n - p - i
\leq (n\omega n)
2(n - p - i), 0 < i < n - p.
Because of 0 < i < n - p, so inequality (1.7) is given.
According to the equality condition of (2.5), we see that equality hold in (1.7) if and only if K
is a ball.
Theorem 1.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
606 X. WAN, W. WANG
Proof of Theorem 1.3. For K,L \in \scrK n
o , since 1 \leq p \leq q, i is a real and 0 \leq i < n, and
\rho n+p - i
K (u)\rho - p
L\ast (u) =
\Bigl[
\rho n+q - i
K (u)\rho - q
L\ast (u)
\Bigr] p/q \bigl[
\rho n - i
K (u)
\bigr] q - p
q ,
then using the Hölder inequality, (2.3) and (2.7) we obtain
\widetilde W - p,i(K,L\ast ) =
1
n
\int
Sn - 1
\rho n+p - i
K (u)\rho - p
L\ast (u)dS(u) =
=
1
n
\int
Sn - 1
\Bigl[
\rho n+q - i
K (u)\rho - q
L\ast (u)
\Bigr] p/q \bigl[
\rho n - i
K (u)
\bigr] q - p
q dS(u) \leq
\leq \widetilde W - q,i(K,L\ast )p/q\widetilde Wi(K)
q - p
q ,
that is \Biggl( \widetilde W - p,i(K,L\ast )\widetilde Wi(K)
\Biggr) 1/p
\leq
\Biggl( \widetilde W - q,i(K,L\ast )\widetilde Wi(K)
\Biggr) 1/q
. (3.3)
The definition of \widetilde \Omega - p,i(K) can be rewritten as
1\widetilde Wi(K)
\Biggl( \widetilde \Omega - p,i(K)
n\widetilde Wi(K)
\Biggr) n - p - i
p
= \mathrm{i}\mathrm{n}\mathrm{f}
\left\{
\Biggl( \widetilde W - p,i(K,Q\ast )\widetilde Wi(K)
\Biggr) n - i
p \widetilde Wi(Q) - 1 : Q \in \scrK n
c
\right\} .
Associated with (3.3) and notice n - i > 0, we can get (1.8).
Theorem 1.3 is proved.
4. Brunn – Minkowski type inequality. In this section, we give Brunn – Minkowski type
inequality for the Lp-dual mixed affine surface areas. First, we prove Theorem 1.4. Next, associated
with the Lp-radial combination of star bodies, we get another Brunn – Minkowski type inequality.
Here the proof of Theorem 1.4 require a lemma as follows:
Lemma 4.1. If K,L \in Sn
o , p \geq 1 and \lambda , \mu \geq 0 (not both zero), real i < n+2p and i \not = n, n+p,
then for any Q \in Sn
o ,\widetilde W - p,i(\lambda \cdot K + - p \mu \cdot L,Q)
- p
n+p - i \geq \lambda \widetilde W - p,i(K,Q)
- p
n+p - i + \mu \widetilde W - p,i(L,Q)
- p
n+p - i (4.1)
with equality if and only if K and L are dilates.
Proof. Since i < n+ 2p and i \not = n, n+ p, thus - (n+ p - i)/p < 0 when i < n+ p and i \not = n,
or 0 < - (n+p - i)/p < 1 when n+p < i < n+2p. Hence by (2.2), (2.7) and Minkowski’s integral
inequality (see [2]), we have
\widetilde W - p,i(\lambda \cdot K + - p \mu \cdot L,Q)
- p
n+p - i =
=
\left[ 1
n
\int
Sn - 1
\rho (\lambda \cdot K + - p \mu \cdot L, u)n+p - i\rho (Q, u) - pdu
\right] - p
n+p - i
=
=
\left[ 1
n
\int
Sn - 1
[\rho (\lambda \cdot K + - p \mu \cdot L, u) - p\rho (Q, u)
p2
n+p - i ]
- n+p - i
p du
\right] - p
n+p - i
=
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
Lp-DUAL MIXED AFFINE SURFACE AREAS 607
=
\left[ 1
n
\int
Sn - 1
[(\lambda \rho (K,u) - p + \mu \rho (L, u) - p)\rho (Q, u)
p2
n+p - i ]
- n+p - i
p du
\right] - p
n+p - i
\geq
\geq \lambda
\left[ 1
n
\int
Sn - 1
\rho (K,u)n+p - i\rho (Q, u) - pdu
\right] - p
n+p - i
+
+\mu
\left[ 1
n
\int
Sn - 1
\rho (L, u)n+p - i\rho (Q, u) - pdu
\right] - p
n+p - i
=
= \lambda \widetilde W - p,i(K,Q)
- p
n+p - i + \mu \widetilde W - p,i(L,Q)
- p
n+p - i for any Q \in Sn
o .
According to the equality condition of Minkowski’s integral inequality, we see that equality holds
in (4.1) if and only if K and L are dilates.
Lemma 4.1 is proved.
Proof of Theorem 1.4. From definition (1.4) and inequality (4.2), we obtain\Bigl[
n
p
n - i \widetilde \Omega - p,i(\lambda \cdot K + - p \mu \cdot L)
\Bigr] - p
n+p - i
=
= \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{ \Bigl[
n\widetilde W - p,i(\lambda \cdot K + - p \mu \cdot L,Q\ast )\widetilde Wi(Q) -
p
n - i
\Bigr] - p
n+p - i
: Q \in \scrK n
c
\biggr\}
=
= \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{ \Bigl[
n\widetilde W - p,i(\lambda \cdot K + - p \mu \cdot L,Q\ast )
\Bigr] - p
n+p - i \widetilde Wi(Q)
p2
(n - i)(n+p - i) : Q \in \scrK n
c
\biggr\}
\geq
\geq \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{ \Bigl[
\lambda (n\widetilde W - p,i(K,Q\ast ))
- p
n+p - i + \mu (n\widetilde W - p,i(L,Q
\ast ))
- p
n+p - i
\Bigr] \widetilde Wi(Q)
p2
(n - i)(n+p - i) : Q \in \scrK n
c
\biggr\}
\geq
\geq \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\lambda
\Bigl[
n\widetilde W - p,i(K,Q\ast )\widetilde Wi(Q) -
p
n - i
\Bigr] - p
n+p - i
: Q \in \scrK n
c
\biggr\}
+
+ \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\mu
\Bigl[
n\widetilde W - p,i(L,Q
\ast )\widetilde Wi(Q) -
p
n - i
\Bigr] - p
n+p - i
: Q \in \scrK n
c
\biggr\}
=
= \lambda
\Bigl[
n - p
n - i \widetilde \Omega - p,i(K)
\Bigr] - p
n+p - i
+ \mu
\Bigl[
n - p
n - i \widetilde \Omega - p,i(L)
\Bigr] - p
n+p - i
.
This yields inequality (1.9).
By the equality condition of (4.1) we know that equality holds in (1.9) if and only if K and L
are dilates.
Theorem 1.4 is proved.
Let i = 0 in Theorem 1.4 and combine with definition (1.2), we have the following corollary.
Corollary 4.1. If K,L \in Sn
o , p \geq 1 and \lambda , \mu \geq 0 (not both zero), then
\widetilde \Omega - p(\lambda \cdot K + - p \mu \cdot L) -
p
n+p \geq \lambda \widetilde \Omega - p(K)
- p
n+p + \mu \widetilde \Omega - p(L)
- p
n+p
with equality if and only if K and L are dilates.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
608 X. WAN, W. WANG
For K,L \in \scrS n
o , p \geq 1 and \lambda , \mu \geq 0 (not both zero), the Lp-radial combination, \lambda \circ K \~+p\mu \circ L \in \scrS n
o ,
of K and L is defined by (see [20])
\rho (\lambda \circ K \~+p\mu \circ L, \cdot )p = \lambda \rho (K, \cdot )p + \mu \rho (L, \cdot )p. (4.2)
According to definition (4.2) of the Lp-radial combination, Wang and He in [14] showed the
Brunn – Minkowski type inequality for the Lp-dual affine surface area as follows:
Theorem 4.A. If K,L \in Kn
c , n > p \geq 1, then
\widetilde \Omega - p(K \~+n+pL)
n - p
n \geq \widetilde \Omega - p(K)
n - p
n + \widetilde \Omega - p(L)
n - p
n (4.3)
with equality if and only if K and L are dilates.
Associated with the Lp-radial combination of star bodies, we establish a Brunn – Minkowski
inequality for the Lp-dual mixed affine surface areas. Under the definition (4.2) of Lp-radial combi-
nation we have the following theorem.
Theorem 4.1. If K,L \in Kn
c , p \geq 1, real i \leq n+ p - 1 and i \not = n, then
\widetilde \Omega - p,i(\lambda \circ K \~+n+p - i\mu \circ L)
n - p - i
n - i \geq \lambda \widetilde \Omega - p,i(K)
n - p - i
n - i + \mu \widetilde \Omega - p,i(L)
n - p - i
n - i (4.4)
with equality if and only if K and L are dilates.
Proof. Since n+ p - i \geq 1 and i \not = n, thus from definition (1.4) and formula (2.7) we have
n
p
n - i \widetilde \Omega - p,i(\lambda \circ K \~+n+p - i\mu \circ L)
n - p - i
n - i =
= \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
n\widetilde W - p,i(\lambda \circ K \~+n+p - i\mu \circ L,Q\ast )\widetilde Wi(Q) -
p
n - i : Q \in \scrK n
c
\Bigr\}
=
= \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
n
\Bigl[
\lambda \widetilde W - p,i(K,Q\ast ) + \mu \widetilde W - p,i(L,Q
\ast )
\Bigr] \widetilde Wi(Q) -
p
n - i : Q \in \scrK n
c
\Bigr\}
=
= \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
n\lambda \widetilde W - p,i(K,Q\ast )\widetilde Wi(Q) -
p
n - i + n\mu \widetilde W - p,i(L,Q
\ast )\widetilde Wi(Q) -
p
n - i : Q \in \scrK n
c
\Bigr\}
\geq
\geq \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
n\lambda \widetilde W - p,i(K,Q\ast )\widetilde Wi(Q) -
p
n - i : Q \in \scrK n
c
\Bigr\}
+
+ \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
n\mu \widetilde W - p,i(L,Q
\ast )\widetilde Wi(Q) -
p
n - i : Q \in \scrK n
c
\Bigr\}
=
= n
p
n - i\lambda \widetilde \Omega - p,i(K)
n - p - i
n - i + n
p
n - i\mu \widetilde \Omega - p,i(L)
n - p - i
n - i .
Thus \widetilde \Omega - p,i(\lambda \circ K \~+n+p - i\mu \circ L)
n - p - i
n - i \geq \lambda \widetilde \Omega - p,i(K)
n - p - i
n - i + \mu \widetilde \Omega - p,i(L)
n - p - i
n - i .
The equality holds if and only if \lambda \circ K \~+n+p - i\mu \circ L are dilates with K and L, respectively. This
mean that equality holds in (4.4) if and only if K and L are dilates.
Theorem 4.1 is proved.
Obviously, by (1.3) we know that if i = 0 and \lambda = \mu = 1 in Theorem 4.1 then inequality (4.4) is
just inequality (4.3).
References
1. Blaschke W. Vorlesungen über differential Geometrie II. Affine Differentialgeometrie. – Berlin: Springer-Verlag,
1923.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
Lp-DUAL MIXED AFFINE SURFACE AREAS 609
2. Gardner R. J. Geometric tomography. – Second ed. – Cambridge: Cambridge Univ. Press, 2006.
3. Ghandehari M. Polar duals of convex bodies // Proc. Amer. Math. Soc. – 1991. – 113. – P. 799 – 808.
4. Hardy G. H., Littlewood J. E., Pólya G. Inequalities. – Cambridge: Cambridge Univ. Press, 1959.
5. K. Leichtwei Zur affinoberfläche konvexer körper // Manuscr. Math. – 1986. – 56. – P. 429 – 464.
6. LeichtweiK. Über einige eigenschaften der affinoberfläche beliebiger konvexer körper // Results Math. – 1988. – 13. –
P. 255 – 282.
7. Ludwig M. General affine surface areas // Adv. Math. – 2010. – 224. – P. 2346 – 2360.
8. M Ludwig, Reitzner M. A classification of SL(n) invariant valuations // Ann. Math. – 2010. – 172. – P. 1219 – 1267.
9. Lutwak E. Dual mixed volumes // Pacif. J. Math. – 1975. – 58. – P. 531 – 538.
10. Lutwak E. Mixed affine surface area // J. Math. Anal. and Appl. – 1987. – 125. – P. 351 – 360.
11. Lutwak E. Centroid bodies and dual mixed volumes // Proc. London Math. Soc. – 1990. – 60, № 2. – P. 365 – 391.
12. Lutwak E. The Brunn – Minkowski – Firey theory I: mixed volumes and the Minkowski problem // J. Differential
Geom. – 1993. – 38, № 1. – P. 131 – 150.
13. Lutwak E. The Brunn – Minkowski – Firey theory II: affine and geominimal surface areas // Adv. Math. – 1996. –
118, № 2. – P. 244 – 294.
14. Wang W., He B. W. Lp-dual affine surface area // J. Math. Anal. and Appl. – 2008. – 348. – P. 746 – 751.
15. Wang W. D., Leng G. S. Lp-dual mixed quermassintegrals // Indian J. Pure Appl. and Math. – 2005. – 36, № 4. –
P. 177 – 188.
16. Wang W. D., Leng G. S. Lp-mixed affine surface area // J. Math. Anal. and Appl. – 2007. – 335, № 1. – P. 341 – 354.
17. Wang W. D., Leng G. S. Some affine isoperimetric inequalities associated with Lp-affine surface area // Houston J.
Math. – 2008. – 34, № 2. – P. 443 – 453.
18. Werner E., Ye D. New Lp-affine isoperimetric inequalities // Adv. Math. – 2008. – 218, № 3. – P. 762 – 780.
19. Werner E., Ye D. Inequalities for mixed p-affine surface area // Math. Ann. – 2010. – 347, № 3. – P. 703 – 737.
20. Schneider R. Convex bodies: the Brunn – Minkowski theory. – Cambridge: Cambridge Univ. Press, 1993.
21. Schütt C., Werner E. Surface bodies and p-affine surface area // Adv. Math. – 2004. – 187. – P. 98 – 145.
Received 25.05.13
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 5
|
| id | umjimathkievua-article-1864 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:08Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/61/693e8ad2c42cdaa436de8235eaccd161.pdf |
| spelling | umjimathkievua-article-18642019-12-05T09:30:15Z $L_p$ -dual mixed affine surface areas $L_p$ -дуальні змiшані афiнні поверхневі площи. Wan, Xiaoyan Wang, W. Ван, Сяоянь Ванг, В. Lutwak proposed the notion of $L_p$-affine surface area according to the Lp-mixed volume. Recently,Wang and He introduced the concept of Lp-dual affine surface area combing with the $L_p$-dual mixed volume. In the article, we give the concept of $L_p$-dual mixed affine surface areas associated with the $L_p$-dual mixed quermassintegrals. Further, some inequalities for the $L_p$-dual mixed affine surface areas are obtained. Лутвок запропонував поняття $L_p$-афiнноЇ поверхневої площi, що вiдповiдає поняттю $L_p$-змiшаного об’єму. Нещодавно Ванг i Хе ввели поняття $L_p$-дуальної афiнної поверхневої площi, пов’язаної з $L_p$-дуальним змiшаним об’ємом. В роботi запропоновано поняття $L_p$-дуальної змiшаної афiнної поверхневої площi, що вiдповiдає $L_p$- дуальним змiшаним квермасiнтегралам. Крiм того, наведено деякi нерiвностi для $L_p$-дуальних змiшаних афiнних поверхневих площ. Institute of Mathematics, NAS of Ukraine 2016-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1864 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 5 (2016); 601-609 Український математичний журнал; Том 68 № 5 (2016); 601-609 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1864/846 Copyright (c) 2016 Wan Xiaoyan; Wang W. |
| spellingShingle | Wan, Xiaoyan Wang, W. Ван, Сяоянь Ванг, В. $L_p$ -dual mixed affine surface areas |
| title | $L_p$ -dual mixed affine surface areas |
| title_alt | $L_p$ -дуальні змiшані афiнні поверхневі площи. |
| title_full | $L_p$ -dual mixed affine surface areas |
| title_fullStr | $L_p$ -dual mixed affine surface areas |
| title_full_unstemmed | $L_p$ -dual mixed affine surface areas |
| title_short | $L_p$ -dual mixed affine surface areas |
| title_sort | $l_p$ -dual mixed affine surface areas |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1864 |
| work_keys_str_mv | AT wanxiaoyan lpdualmixedaffinesurfaceareas AT wangw lpdualmixedaffinesurfaceareas AT vansâoânʹ lpdualmixedaffinesurfaceareas AT vangv lpdualmixedaffinesurfaceareas AT wanxiaoyan lpdualʹnízmišaníafinnípoverhnevíploŝi AT wangw lpdualʹnízmišaníafinnípoverhnevíploŝi AT vansâoânʹ lpdualʹnízmišaníafinnípoverhnevíploŝi AT vangv lpdualʹnízmišaníafinnípoverhnevíploŝi |