陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Open question: the Mahler conjecture on convex bodies」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

It seems that I have unwittingly started an “open problem of the week” column here; certainly it seems easier for me to pose unsolved problems than to write papers 🙂 .

This question in convex geometry has been around for a while; I am fond of it because it attempts to capture the intuitively obvious fact that cubes and octahedra are the “pointiest” possible symmetric convex bodies one can create. Sadly, we still have very few tools to make this intuition rigorous (especially when compared against the assertion that the Euclidean ball is the “roundest” possible convex body, for which we have many rigorous and useful formulations).

已知结果和反例

To state the conjecture I need a little notation. Suppose we have a symmetric convex body in a Euclidean space, thus B is open, convex, bounded, and symmetric around the origin. We can define the polar body by

This is another symmetric convex body. One can interpret B as the unit ball of a Banach space norm on , in which case is simply the unit ball of the dual norm. The Mahler volume of the body is defined as the product of the volumes of B and its polar body:

证明或构造的主线

One feature of this Mahler volume is that it is an affine invariant: if is any invertible linear transformation, then TB has the same Mahler volume as B. It is also clear that a body has the same Mahler volume as its polar body. Finally the Mahler volume reacts well to Cartesian products: if are convex bodies, one can check that

For the unit Euclidean ball one can easily compute the Mahler volume as

阅读时建议盯住的点

while for the unit cube or the unit octahedron the Mahler volume is

One can also think of as the unit balls of the norms respectively.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源