{"paper":{"title":"The structure of continuous rigid functions of two variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"M\\'arton Elekes, Rich\\'ard Balka","submitted_at":"2011-09-22T20:11:11Z","abstract_excerpt":"A function $f:\\RR^n \\to \\RR$ is called \\emph{vertically rigid} if $graph(cf)$ is isometric to $graph (f)$ for all $c \\neq 0$. We settled Jankovi\\'c's conjecture in a separate paper by showing that a continuous function $f:\\RR \\to \\RR$ is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \\in \\RR$). Now we prove that a continuous function $f:\\RR^2 \\to \\RR$ is vertically rigid if and only if after a suitable rotation around the z-axis $f(x,y)$ is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ ($a,b,d,k \\in \\RR$, $k \\neq 0$, $s : \\RR \\to \\RR$ contin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4951","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}