Consider the problem of estimating the volume of a tumor, given X-ray scans along orthogonal axes. It may be known that the tumor has a somewhat spherical shape. To formalise this idea, let be the tumor, the area of its surface, its volume and . From the isoperimetric inequality, we have , with equality iff is a ball. Correspondingly, we say that is a quasi-ball if . In reality, is unknown but its distribution may be determined.
We are now given the areas , and of the projections of along orthogonal axes. From the Loomis-Whitney inequality (or Cauchy-Schwarz in this case), we have the following estimate of the volume of .
Can we find such an estimate of that is close to sharp when is close to a ball?
Loomis, L. H.; Whitney, H. An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55 (1949), no. 10, 961–962. http://projecteuclid.org/euclid.bams/1183514163.