What question did this study set out to answer?

This research aims to extend the ham-sandwich theorem under conditions not requiring a halving of measures.

March 8, 2026Open Access

Weak extension of the ham-sandwich theorem : some ratio not in half

Key Points

This research aims to extend the ham-sandwich theorem under conditions not requiring a halving of measures.
Application of Borsuk's antipodal theorem
Use of measurable sets with positive Lebesgue measure
Introduction of a hyperplane for division
Successful extension of the ham-sandwich theorem to ratios not in half
Demonstrates new conditions for dividing measurable sets

Abstract

The ham-sandwich theorem is famous as an application of Borsuk's antipodal theorem. It states that given n measurable sets A₁,. . . , Aₙ with positive Lebesgue measure in Rⁿ, it is possible to divide each one of them in half by a hyperplane. Recently the author applied Borsuk's antipodal theorem to optimization theory in 3. In this paper we extend the ham-sandwich theorem to some ratio not in half by using the technique of 3.

Weak extension of the ham-sandwich theorem : some ratio not in half

Key Points

Abstract

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