Make your strongest attempt to resolve the following self-contained mathematical problem in this response.
Do not browse the web, retrieve external sources, search previous conversations or connected files, or consult other AI agents. In particular, do not search for the problem’s name, publication history, or known status. You may use built-in code execution, if available, for your own calculations and experiments, without network access.
If you already recognize the exact problem or recall a solution, briefly disclose what you recognize before proceeding. Still supply the mathematical argument rather than merely citing a remembered result.
Let d ≥ 2 be an integer. A convex body K in R^d is a compact convex set with nonempty interior. Write vol_j for j-dimensional Euclidean volume.