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Let &#x003a3; be a domain on an m-dimensional minimal submanifold in the outside of a convex set C in &#x1D54A;n or &#x210D;n. The modified volume M(&#x003a3;) is introduced by Choe and Gulliver (1992) and we prove a sharp modified relative isoperimetric inequality for the domain &#x003a3;, (1/2)mm&#x003c9;mM(&#x003a3;)m-1&#x02264;Volume(&#x02202;&#x003a3;-&#x02202;C)m, where &#x003c9;m is the volume of the unit ball of &#x211D;m. For any domain &#x003a3; on a minimal surface in the outside convex set C in an n-dimensional Riemannian manifold, we prove a weak relative isoperimetric inequality &#x003c0;Area(&#x003a3;)&#x02264;Length(&#x02202;&#x003a3;-&#x02202;C)2+KArea(&#x003a3;)2, where K is an upper bound of sectional curvature of the Riemannian manifold

Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold