Carbery proved that if \(u:\mathbb {R}^{n} \rightarrow \mathbb {R}\) is a positive, strictly convex function satisfying \(\det D^{2}u \geq 1\), then we have the estimate

and this is optimal. We give a short proof that also implies other results. Our main result is an estimate for the sublevel set of functions \(u:[0,1]^{2} \rightarrow \mathbb {R}\) satisfying 1 ≤Δu ≤ c for some universal constant c: for any α > 0, we have

For ‘typical’ functions, we expect the integral to be finite for α < 1. While Carbery-Christ-Wright have shown that no sublevel set estimates independently of u exist, this result shows that for ‘typical’ functions satisfying \({\Delta } u \sim 1\), we expect the sublevel set to be \(\lesssim \varepsilon ^{1/2-}\). We do not know whether this is sharp or whether similar statements are true in higher dimensions.

Carbery证明，如果\（u：\ mathbb {R} ^ {n} \ rightarrow \ mathbb {R} \）是一个满足\（\ det D ^ {2} u \ geq 1 \）的正，严格凸函数，然后我们有估计

这是最佳的。我们给出一个简短的证明，它也暗示了其他结果。我们的主要结果是针对分段的功能集的估计\（U：[0,1] ^ {2} \ RIGHTARROW \ mathbb {R} \）满足l≤Δ ü ≤ Ç一些通用常量Ç：对于任何α > 0，我们有

对于“典型”函数，我们期望α <1的积分是有限的。虽然Carbery-Christ-Wright已证明不存在独立于u的子集估计，但该结果表明，对于满足\（{ Delta} u \ sim 1 \），我们期望子级集为\（\ lesssim \ varepsilon ^ {1/2-} \）。我们不知道这是不是很尖锐，或者在更高维度上类似的说法是否正确。