Loomis-Whitney type inequality for quasi-balls?

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 T \subset \mathbb{R}^3 be the tumor, s the area of its surface, m its volume and C = s^3/m^2. From the isoperimetric inequality, we have C \geq 6^2 \pi, with equality iff T is a ball. Correspondingly, we say that T is a quasi-ball if C \approx 6^2 \pi. In reality, C is unknown but its distribution may be determined.

We are now given the areas m_1, m_2 and m_3 of the projections of T along orthogonal axes. From the Loomis-Whitney inequality (or Cauchy-Schwarz in this case), we have the following estimate of the volume m of T.

Theorem.
We have

\max_i \sqrt{\frac{2^3 m_i^3}{C}} \le m \le \sqrt{m_1 m_2 m_3}.

Problem.
Can we find such an estimate of m that is close to sharp when T is close to a ball?

References.
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.

Leave a comment

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s