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Abstract While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in$$\mathbb {R}^n$$ $R^n$. In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension$$n=1$$ $n=1$. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition$$(\mathcal {G},\beta )$$ $(G,\beta )$and nonempty sets$$A_1,\dots ,A_m\subseteq \mathbb {R}$$ $A_1,\cdots ,A_m\subseteq R$, equality holds iff for each$$S\in \mathcal {G}$$ $S\in G$, the set$$\sum _{i\in S}A_i$$ ${\sum }_{i\in S}{A}_i$is an interval. In the case of dimension$$n\ge 2$$ $n\ge 2$we will show that equality can hold if and only if the set$$\sum _{i=1}^{m}A_i$$ ${\sum }_{i=1}^m{A}_i$has measure 0. 

Path guiding is a promising technique to reduce the variance of path tracing. Although existing online path guiding algorithms can eventually learn good sampling distributions given a large amount of time and samples, the speed of learning becomes a major bottleneck. In this paper, we accelerate the learning of sampling distributions by training a light-weight neural network offline to reconstruct from sparse samples. Uniquely, we design our neural network to directly operate convolutions on a sparse quadtree, which regresses a high-quality hierarchical sampling distribution. Our approach can reconstruct reasonably accurate sampling distributions faster, allowing for efficient path guiding and rendering. In contrast to the recent offline neural path guiding techniques that reconstruct low-resolution 2D images for sampling, our novel hierarchical framework enables more fine-grained directional sampling with less memory usage, effectively advancing the practicality and efficiency of neural path guiding. In addition, we take advantage of hybrid bidirectional samples including both path samples and photons, as we have found this more robust to different light transport scenarios compared to using only one type of sample as in previous work. Experiments on diverse testing scenes demonstrate that our approach often improves rendering results with better visual quality and lower errors. Our framework can also provide the proper balance of speed, memory cost, and robustness.