Abstract
Motivated by the celebrated work of Lutwak, Yang and Zhang [1] on
1 Introduction
Let
If
where
Let
If
For
where
Based on the
where
Theorem 1.A
If
with equality if and only if K is pselfminimal.
Theorem 1.B
If
with equality if and only if K is pselfminimal.
Very recently, Lutwak et al. [1] introduced the
for each continuous
Using the
Meanwhile, Lutwak et al. [1] gave the following formula of
In [1], Lutwak et al. introduced the
The case
By (1.2), the following special cases are showed:
The
Based on the
Definition 1.1
For
If
According to the definition of
Theorem 1.C
Suppose
with equality if and only if K and L are dilates.
Obviously, Theorem 1.C for
In addition, they [2] gave a BrunnMinkowskitype inequality for the
Theorem 1.D
Suppose
with equality if and only if
2 Main results
Here,
In this article, associated with the
Theorem 1.1
Suppose
with equality if and only if K and L are dilates.
Theorem 1.2
Suppose
with equality if and only if K and L are dilates.
Remark 1.1
When
Next, we obtain an affine isoperimetric inequality for the
Theorem 1.3
Suppose
for
Remark 1.2
If
Finally, together with the
Theorem 1.4
Suppose
with equality if and only if
Here,
3 Background
In order to complete the proofs of Theorems 1.1–1.4, we collect some basic facts about convex bodies and star bodies.
If
Meanwhile, it is easy to get that
For
where the operation “
For
where the operation “
4 Proofs of main theorems
In this section, we will prove Theorems 1.1–1.4. To complete the proof of Theorem 1.1, we require the following lemma.
Lemma 3.1
[37] Suppose
with equality if and only if K and L are dilates.
Proof of Theorem 1.1
Together (1.6) with inequality (3.1), we obtain that for
This is
This yields inequality (1.7). According to the equality condition of inequality (3.1), we see that the equality of the above inequality holds if and only if
Lemma 3.2
[37] Suppose
with equality if and only if K and L are dilates.
Proof of Theorem 1.2
From Definition 1.1 and (3.2) we see that for
This is, for
This gives inequality (1.8). From the equality condition of inequality (3.2), we see that the equality holds in (1.8) if and only if
Lemma 3.3
[2] If
For
Proof of Theorem 1.3
Since
Taking
i.e.,
By the equality condition of inequality (3.3) and Definition 1.1, for
Lemma 3.4
Suppose
with equality if and only if
Proof
Since
This yields inequality (3.5).
By the equality condition of Minkowski’s integral inequality, we see that equality holds in (3.5) if and only if
Proof of Theorem 1.4
Because of
i.e.,
This gives inequality (1.10).
According to the equality condition of inequality (3.5), we see that the equality holds in (1.10) if and only if
Acknowledgments
The authors would like to express their cordial gratitude to the referee for valuable comments which improved the paper.

Funding information: This work was supported by the NNSF of China (Grant Nos 11871275 and 11371194).

Conflict of interest: The authors state no conflict of interest.
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