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Ein Übertragungsprinzip für konvexe Körper

Kurt Mahler

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Journal citation: Časopis pro pěstování matematiky a fysiky 68(3), 93–102 (1939)
Publication date: 1939
DOI: 10.21136/CPMF.1939.109441 ↗

Role in dependence graphs

Proof-critical source

Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle

This paper is included only for the following marked statement:

Consecutive-minima transference theorem · pp. 93–102; restated as German–Evdokimov Theorem DRelates successive minima of a parallelepiped and its pseudo-compound dual.

Proof-critical source

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

This paper is included only for the following marked statement:

Section 3 transference theorem · journal pp. 99–101Compares successive minima of a convex body and its compound body.

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Version of record · Časopis pro pěstování matematiky a fysiky 68 (1939), 93–102

Kurt Mahler. Ein Übertragungsprinzip für konvexe Körper. Časopis pro pěstování matematiky a fysiky 68 (1939), no. 3–4, 93–102.

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Generated September 26, 2026

01Statements2 reported findingsNo statement issues found

The consecutive-minima transference theorem for a centrally symmetric convex body and its polar is correct. In particular, if $\sigma^{(h)}$ and $\tau^{(h)}$ are their successive minima, then $1\leq\sigma^{(h)}\tau^{(n-h+1)}\leq(n!)^2$ for every $h$ .

Principal transference theoremNo issue found

Each paired pair of successive minima has the claimed dimension-only bounds

Journal pages 99–101 · equations (A) and (B) · version of record

The lower bound follows because the $h$ selected lattice vectors for the body and the $n-h+1$ selected lattice vectors for the polar cannot span orthogonal subspaces: their dimensions sum to $n+1$ . Hence one scalar product is a nonzero integer, so polarity gives $1\leq\sigma^{(h)}\tau^{(n-h+1)}$ . Multiplying all paired factors and using the product bounds for the two sequences of successive minima gives the upper bound $(n!)^2$ . The argument has the correct index reversal and covers every $1\leq h\leq n$ .

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Inhomogeneous consequenceNo issue found

The final inhomogeneous bound follows from the transference estimate

Journal pages 101–102 · §4, equations (15)–(16) · version of record

The previously established inhomogeneous covering estimate is combined with equation (B) to replace the last minimum of the primal body by the first minimum of the polar body, up to a constant depending only on dimension. The accompanying measure estimate correctly explains why the resulting exponent cannot be uniformly improved apart from such a constant.

02Proofs2 reported findingsNo proof issues found

The polar-volume comparison, product bounds for successive minima, and nonorthogonality argument form a complete proof once the explicitly cited classical Minkowski product inequalities are taken as the terminal input.

Polar-volume comparisonNo issue found in the reviewed proof

The parallelepiped–cross-polytope sandwich yields the required volume-product bounds

Journal pages 96–98 · §2, equations (6)–(9) · version of record

Supporting hyperplanes and their contact points construct an outer parallelepiped and an inner cross-polytope for each polar body. Their volumes differ by the factor $n!$ , while the exact product for a polar parallelepiped and cross-polytope is $4^n/n!$ . The two containments therefore give the two-sided estimate for $JJ'$ used in §3.

Classical terminal inputCorrectly sourced terminal input

The only imported proof-critical inequalities are identified

Journal page 99 · equations (10) and (10-prime) · citation to Minkowski, Geometrie der Zahlen

Mahler explicitly invokes Minkowski's classical product inequalities for successive minima and volume. Substituting these inequalities for the body and its polar, together with the independently proved volume-product estimate, gives equation (11) with no missing algebraic step.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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