| 摘要: |
| 对数凹函数嫡的 $L_p$ Shephard 问题是一类摘的比较问题: 若 ${\Pi}_{p}f \leq {\Pi}_{p}g$, 是否当1≤p＜n时，有$\operatorname{Ent}(f) \geqslant \operatorname{Ent}(g)$；当n＜p时，有$\operatorname{Ent}(f) \leqslant \operatorname{Ent}(g)$，其中，$\Pi_p f$为对数凹函数f的$L_p$投影体，得到了该问题的部分解答。 |
| 关键词: $L_p$ Shephard问题 对数凹函数 $L_p$投影体 熵 |
| DOI：10.20192/j.cnki.JSHNU（NS）.2024.05.001 |
| 分类号:O186.5 |
| 基金项目:The National Natural Science Foundation of China (11701373);The Shanghai Sailing Program (17YF1413800) |
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| The Lp Shephard problem on entropy of log-concave functions |
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GAO Tian, LI Shuqian, MA Dan
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Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
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| Abstract: |
| In this paper, we introduce the ${L_p}$ Shephard problem on entropy of log-concave functions, a comparison problem: whether ${\Pi}_{p}f \leq {\Pi}_{p}g$ implies that $Ent(f)\ge Ent(g), \; \mathrm{for} \;1 \le p < n$, and $Ent(f)\leq Ent(g), \;\mathrm{for} \; n < p$, where ${\Pi}_{p}f$ is the ${L_p}$ projection body of a log-concave function $f$. Our results give a partial answer to this problem. |
| Key words: the ${L_p}$ Shephard problem log-concave functions ${L_p}$ projection bodies entropy |
