The $ L_p $-Petty projection inequality for $ s $-concave functions

Tian Gao; Dan Ma; Tian Gao; Dan Ma

doi:10.3934/math.2025353

AIMS Mathematics

2025, Volume 10, Issue 4: 7706-7716. doi: 10.3934/math.2025353

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The $ L_p $-Petty projection inequality for $ s $-concave functions

Tian Gao ,
Dan Ma ^,

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

Received: 06 January 2025 Revised: 24 March 2025 Accepted: 26 March 2025 Published: 01 April 2025
MSC : 26D10, 52A40

We study the "functionalization" of the $ L_p $-projection body and two related important inequalities in geometry. On the class of $ s $-concave functions, a general function counterpart of the $ L_p $-projection body is introduced. In addition, the $ L_p $-Petty projection inequality and the $ L_p $ isoperimetric inequality are established on this class. Finally, we show that the $ L_p $-Petty projection inequality strengthens the $ L_p $ isoperimetric inequality on the same class.
- $ s $-concave function,
- $ L_p $-projection body,
- $ L_p $-Petty projection inequality,
- $ L_p $ isoperimetric inequality
Citation: Tian Gao, Dan Ma. The $ L_p $-Petty projection inequality for $ s $-concave functions[J]. AIMS Mathematics, 2025, 10(4): 7706-7716. doi: 10.3934/math.2025353

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Abstract

We study the "functionalization" of the $ L_p $-projection body and two related important inequalities in geometry. On the class of $ s $-concave functions, a general function counterpart of the $ L_p $-projection body is introduced. In addition, the $ L_p $-Petty projection inequality and the $ L_p $ isoperimetric inequality are established on this class. Finally, we show that the $ L_p $-Petty projection inequality strengthens the $ L_p $ isoperimetric inequality on the same class.

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