Last geometric statement of Jacobi

In differential geometry, the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi, which states:

Every caustic from any point $p$ on an ellipsoid other than umbilical points has exactly four cusps.{{citation | last = Arnold | first = V. I. | authorlink = Vladimir Arnold | contribution = Topological problems in wave propagation theory and topological economy principle in algebraic geometry | location = Providence, RI | mr = 1733567 | pages = 39–54 | publisher = Amer. Math. Soc. | series = Fields Inst. Commun. | title = The Arnoldfest (Toronto, ON, 1997) | volume = 24 | year = 1999}}

Numerical experiments had indicated the statement is true{{cite journal | last = Sinclair | first = R. | issue = 4 | journal = Experimental Mathematics | mr = 2043997 | pages = 477–485 | title = On the last geometric statement of Jacobi | url = http://projecteuclid.org/euclid.em/1087568023 | volume = 12 | year = 2003 | doi=10.1080/10586458.2003.10504515| s2cid = 13520470 }} before it was proven rigorously in 2004 by Itoh and Kiyohara.{{cite journal |first1=J. |last1= Itoh |first2=K. |last2= Kiyohara |year=2004 |title= The cut loci and the conjugate loci on ellipsoids |journal= Manuscripta Mathematica |volume=114 |issue=2 |pages=247–264 |doi= 10.1007/s00229-004-0455-z |s2cid= 121131543 }} It has since been extended to a wider class of surfaces beyond the ellipsoid.{{cite journal | last1 = Sinclair | first1 = R. | last2 = Tanaka | first2 = M. | doi = 10.1090/S0025-5718-06-01924-7 | issue = 256 | journal = Mathematics of Computation | mr = 2240635 | pages = 1779–1808 | title = Jacobi's last geometric statement extends to a wider class of Liouville surfaces | volume = 75 | year = 2006| bibcode = 2006MaCom..75.1779S | doi-access = free}}