α-凹函数的α-卷积的积分不等式
Integral inequalities for $α$-convolutions of $α$-concave functions
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中文总结 AI 辅助
该研究针对α-凹函数的α-卷积，建立了Plünnecke–Ruzsa型不等式等积分不等式，衔接了对数凹与拟凹相关研究，丰富了凸分析领域的积分不等式理论。
中文摘要 AI 辅助
源自加法组合学的经典和集不等式，在有限维实向量空间的凸体上存在几何类似物，这是Fradelizi与本文两位作者近期的研究成果。我们针对几何α-凹函数在α-卷积下的情况，发展了积分类似物，α-卷积是“概率几何化”方案中自然产生的一种运算。具体而言，我们为α-凹函数的α-卷积建立了Plünnecke–Ruzsa型不等式，以及和差不等式、Ruzsa三角不等式的精确类似物；还证明了α-凹函数的精确Rogers–Shephard型不等式，并刻画了其等号成立的情形，该结果衔接了Alonso-Gutiérrez等人研究的对数凹情形与Colesanti研究的拟凹情形。
英文摘要
Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $α$-concave functions under $α$-convolutions, an operation that arises naturally in the ``geometrization of probability'' program. In particular, we establish a Plünnecke--Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $α$-convolutions of $α$-concave functions. We also prove a sharp Rogers--Shephard-type inequality and characterize its equality cases for $α$-concave functions, bridging the log-concave case studied by Alonso-Gutiérrez, González-Merino, Jiménez, and Villa and the quasi-concave case studied by Colesanti.
