Hadwiger’s illumination conjecture
Can every convex body in n dimensions have its entire boundary illuminated from at most 2ⁿ directions?
Convex geometry
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Symmetric Mahler conjecture
Do cubes and their dual shapes minimize a symmetric body’s volume multiplied by its polar’s volume?
Moser’s convex worm problem
What is the smallest-area convex region that can contain every planar curve of length one after repositioning it?
Kannan–Lovász–Simonovits conjecture
Can local changes control the fluctuations of functions under log-concave probability laws with a constant independent of dimension?